Practice · A half turn hides the overlap twice over · Sensim Math

Variant 1 medium answer: 4 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $124\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 22 cm by 13 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 124 cm. We need how far O is from C.

Givens

The rectangle is 22 cm by 13 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 124 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 22 cm by 13 cm, so one rectangle's perimeter is 2 times 22 plus 2 times 13.

2 \times 22 + 2 \times 13 = 44 + 26 = 70 \text{ cm}

70 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

70 + 70 - 2 \times (2 \times OC) = 140 - 4 \times OC

The only unknown left is OC.

5Work backwards from 124 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 124 cm, so 140 minus 4 times OC equals 124. That means 4 times OC equals 16, so OC equals 4.

140 - 4 \times OC = 124 \;\Rightarrow\; 4 \times OC = 16 \;\Rightarrow\; OC = 4 \text{ cm}

O sits 4 cm above C.

Answer: 4 cm

4 · Reviewdoes it hold up?

Put 4 back in: 140 - 4 x 4 = 140 - 16 = 124 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 13 / 2 = 6.5 cm, because O is nearer C than D. 4 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 2 medium answer: 1 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $76\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 14 cm by 6 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 76 cm. We need how far O is from C.

Givens

The rectangle is 14 cm by 6 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 76 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 14 cm by 6 cm, so one rectangle's perimeter is 2 times 14 plus 2 times 6.

2 \times 14 + 2 \times 6 = 28 + 12 = 40 \text{ cm}

40 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

40 + 40 - 2 \times (2 \times OC) = 80 - 4 \times OC

The only unknown left is OC.

5Work backwards from 76 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 76 cm, so 80 minus 4 times OC equals 76. That means 4 times OC equals 4, so OC equals 1.

80 - 4 \times OC = 76 \;\Rightarrow\; 4 \times OC = 4 \;\Rightarrow\; OC = 1 \text{ cm}

O sits 1 cm above C.

Answer: 1 cm

4 · Reviewdoes it hold up?

Put 1 back in: 80 - 4 x 1 = 80 - 4 = 76 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 6 / 2 = 3 cm, because O is nearer C than D. 1 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 3 medium answer: 8 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $140\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 25 cm by 18 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 140 cm. We need how far O is from C.

Givens

The rectangle is 25 cm by 18 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 140 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 25 cm by 18 cm, so one rectangle's perimeter is 2 times 25 plus 2 times 18.

2 \times 25 + 2 \times 18 = 50 + 36 = 86 \text{ cm}

86 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

86 + 86 - 2 \times (2 \times OC) = 172 - 4 \times OC

The only unknown left is OC.

5Work backwards from 140 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 140 cm, so 172 minus 4 times OC equals 140. That means 4 times OC equals 32, so OC equals 8.

172 - 4 \times OC = 140 \;\Rightarrow\; 4 \times OC = 32 \;\Rightarrow\; OC = 8 \text{ cm}

O sits 8 cm above C.

Answer: 8 cm

4 · Reviewdoes it hold up?

Put 8 back in: 172 - 4 x 8 = 172 - 32 = 140 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 18 / 2 = 9 cm, because O is nearer C than D. 8 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 4 medium answer: 9 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $156\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 28 cm by 20 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 156 cm. We need how far O is from C.

Givens

The rectangle is 28 cm by 20 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 156 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 28 cm by 20 cm, so one rectangle's perimeter is 2 times 28 plus 2 times 20.

2 \times 28 + 2 \times 20 = 56 + 40 = 96 \text{ cm}

96 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

96 + 96 - 2 \times (2 \times OC) = 192 - 4 \times OC

The only unknown left is OC.

5Work backwards from 156 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 156 cm, so 192 minus 4 times OC equals 156. That means 4 times OC equals 36, so OC equals 9.

192 - 4 \times OC = 156 \;\Rightarrow\; 4 \times OC = 36 \;\Rightarrow\; OC = 9 \text{ cm}

O sits 9 cm above C.

Answer: 9 cm

4 · Reviewdoes it hold up?

Put 9 back in: 192 - 4 x 9 = 192 - 36 = 156 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 20 / 2 = 10 cm, because O is nearer C than D. 9 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 5 medium answer: 3 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $84\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 15 cm by 9 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 84 cm. We need how far O is from C.

Givens

The rectangle is 15 cm by 9 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 84 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 15 cm by 9 cm, so one rectangle's perimeter is 2 times 15 plus 2 times 9.

2 \times 15 + 2 \times 9 = 30 + 18 = 48 \text{ cm}

48 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

48 + 48 - 2 \times (2 \times OC) = 96 - 4 \times OC

The only unknown left is OC.

5Work backwards from 84 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 84 cm, so 96 minus 4 times OC equals 84. That means 4 times OC equals 12, so OC equals 3.

96 - 4 \times OC = 84 \;\Rightarrow\; 4 \times OC = 12 \;\Rightarrow\; OC = 3 \text{ cm}

O sits 3 cm above C.

Answer: 3 cm

4 · Reviewdoes it hold up?

Put 3 back in: 96 - 4 x 3 = 96 - 12 = 84 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 9 / 2 = 4.5 cm, because O is nearer C than D. 3 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 6 medium answer: 4 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $96\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 18 cm by 10 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 96 cm. We need how far O is from C.

Givens

The rectangle is 18 cm by 10 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 96 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 18 cm by 10 cm, so one rectangle's perimeter is 2 times 18 plus 2 times 10.

2 \times 18 + 2 \times 10 = 36 + 20 = 56 \text{ cm}

56 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

56 + 56 - 2 \times (2 \times OC) = 112 - 4 \times OC

The only unknown left is OC.

5Work backwards from 96 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 96 cm, so 112 minus 4 times OC equals 96. That means 4 times OC equals 16, so OC equals 4.

112 - 4 \times OC = 96 \;\Rightarrow\; 4 \times OC = 16 \;\Rightarrow\; OC = 4 \text{ cm}

O sits 4 cm above C.

Answer: 4 cm

4 · Reviewdoes it hold up?

Put 4 back in: 112 - 4 x 4 = 112 - 16 = 96 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 10 / 2 = 5 cm, because O is nearer C than D. 4 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 7 medium answer: 7 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $156\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 30 cm by 16 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 156 cm. We need how far O is from C.

Givens

The rectangle is 30 cm by 16 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 156 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 30 cm by 16 cm, so one rectangle's perimeter is 2 times 30 plus 2 times 16.

2 \times 30 + 2 \times 16 = 60 + 32 = 92 \text{ cm}

92 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

92 + 92 - 2 \times (2 \times OC) = 184 - 4 \times OC

The only unknown left is OC.

5Work backwards from 156 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 156 cm, so 184 minus 4 times OC equals 156. That means 4 times OC equals 28, so OC equals 7.

184 - 4 \times OC = 156 \;\Rightarrow\; 4 \times OC = 28 \;\Rightarrow\; OC = 7 \text{ cm}

O sits 7 cm above C.

Answer: 7 cm

4 · Reviewdoes it hold up?

Put 7 back in: 184 - 4 x 7 = 184 - 28 = 156 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 16 / 2 = 8 cm, because O is nearer C than D. 7 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 8 medium answer: 3 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $56\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 10 cm by 7 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 56 cm. We need how far O is from C.

Givens

The rectangle is 10 cm by 7 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 56 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 10 cm by 7 cm, so one rectangle's perimeter is 2 times 10 plus 2 times 7.

2 \times 10 + 2 \times 7 = 20 + 14 = 34 \text{ cm}

34 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

34 + 34 - 2 \times (2 \times OC) = 68 - 4 \times OC

The only unknown left is OC.

5Work backwards from 56 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 56 cm, so 68 minus 4 times OC equals 56. That means 4 times OC equals 12, so OC equals 3.

68 - 4 \times OC = 56 \;\Rightarrow\; 4 \times OC = 12 \;\Rightarrow\; OC = 3 \text{ cm}

O sits 3 cm above C.

Answer: 3 cm

4 · Reviewdoes it hold up?

Put 3 back in: 68 - 4 x 3 = 68 - 12 = 56 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 7 / 2 = 3.5 cm, because O is nearer C than D. 3 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 9 medium answer: 5 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $88\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 16 cm by 11 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 88 cm. We need how far O is from C.

Givens

The rectangle is 16 cm by 11 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 88 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 16 cm by 11 cm, so one rectangle's perimeter is 2 times 16 plus 2 times 11.

2 \times 16 + 2 \times 11 = 32 + 22 = 54 \text{ cm}

54 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

54 + 54 - 2 \times (2 \times OC) = 108 - 4 \times OC

The only unknown left is OC.

5Work backwards from 88 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 88 cm, so 108 minus 4 times OC equals 88. That means 4 times OC equals 20, so OC equals 5.

108 - 4 \times OC = 88 \;\Rightarrow\; 4 \times OC = 20 \;\Rightarrow\; OC = 5 \text{ cm}

O sits 5 cm above C.

Answer: 5 cm

4 · Reviewdoes it hold up?

Put 5 back in: 108 - 4 x 5 = 108 - 20 = 88 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 11 / 2 = 5.5 cm, because O is nearer C than D. 5 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 10 medium answer: 2 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $72\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 12 cm by 8 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 72 cm. We need how far O is from C.

Givens

The rectangle is 12 cm by 8 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 72 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 12 cm by 8 cm, so one rectangle's perimeter is 2 times 12 plus 2 times 8.

2 \times 12 + 2 \times 8 = 24 + 16 = 40 \text{ cm}

40 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

40 + 40 - 2 \times (2 \times OC) = 80 - 4 \times OC

The only unknown left is OC.

5Work backwards from 72 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 72 cm, so 80 minus 4 times OC equals 72. That means 4 times OC equals 8, so OC equals 2.

80 - 4 \times OC = 72 \;\Rightarrow\; 4 \times OC = 8 \;\Rightarrow\; OC = 2 \text{ cm}

O sits 2 cm above C.

Answer: 2 cm

4 · Reviewdoes it hold up?

Put 2 back in: 80 - 4 x 2 = 80 - 8 = 72 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 8 / 2 = 4 cm, because O is nearer C than D. 2 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 11 medium answer: 6 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $128\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 24 cm by 14 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 128 cm. We need how far O is from C.

Givens

The rectangle is 24 cm by 14 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 128 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 24 cm by 14 cm, so one rectangle's perimeter is 2 times 24 plus 2 times 14.

2 \times 24 + 2 \times 14 = 48 + 28 = 76 \text{ cm}

76 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

76 + 76 - 2 \times (2 \times OC) = 152 - 4 \times OC

The only unknown left is OC.

5Work backwards from 128 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 128 cm, so 152 minus 4 times OC equals 128. That means 4 times OC equals 24, so OC equals 6.

152 - 4 \times OC = 128 \;\Rightarrow\; 4 \times OC = 24 \;\Rightarrow\; OC = 6 \text{ cm}

O sits 6 cm above C.

Answer: 6 cm

4 · Reviewdoes it hold up?

Put 6 back in: 152 - 4 x 6 = 152 - 24 = 128 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 14 / 2 = 7 cm, because O is nearer C than D. 6 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

Variant 12 medium answer: 5 cm

Rectangle ABCD is rotated $180\degree$ about point O as the center of symmetry to complete a point-symmetric figure. The perimeter of the completed figure is $108\ \text{cm}$ . Find the length of segment OC in centimeters.

Show solution

1 · Understandwhat's really being asked

A 20 cm by 12 cm rectangle is turned a half turn about a point O on its right side, making a point-symmetric figure whose perimeter is 108 cm. We need how far O is from C.

Givens

The rectangle is 20 cm by 12 cm.
O lies on side DC, closer to C.
The completed figure has perimeter 108 cm.

Unknowns

The length of OC.

Constraints

The two rectangles overlap along the line through DC, and the shared strip is inside the finished figure, not on its edge.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#11 Work Backwards

Draw both rectangles and see what stops being on the outside. Write the finished perimeter in terms of OC, then run the arithmetic backwards from the number given.

3 · Execute5 carry out the plan

1Draw the completed figure

#1 Draw a Diagram 4.G.A.3

Draw rectangle ABCD, mark O on the right side DC near C, then draw the copy you get by turning the rectangle a half turn about O. The two rectangles sit on opposite sides of O and share a strip along the line through DC.

Two congruent rectangles, joined along a strip.

2Find the perimeter of one rectangle

#7 Identify Subproblems 4.MD.A.3

Each rectangle is 20 cm by 12 cm, so one rectangle's perimeter is 2 times 20 plus 2 times 12.

2 \times 20 + 2 \times 12 = 40 + 24 = 64 \text{ cm}

64 cm around one of them.

3See which part disappears inside

#1 Draw a Diagram 4.G.A.3

Where the rectangle and its turned copy meet, they share a strip centred on O. It reaches OC below O and the same distance above, so the strip is 2 x OC long -- and it is inside the finished figure, so neither rectangle's copy of it counts.

\text{overlap} = 2 \times OC

Half of it comes from each rectangle.

4Write the perimeter of the whole figure

#7 Identify Subproblems 3.MD.D.8

Add the two rectangle perimeters, then take the overlap off once for each rectangle, because each gave up that shared strip.

64 + 64 - 2 \times (2 \times OC) = 128 - 4 \times OC

The only unknown left is OC.

5Work backwards from 108 cm

#11 Work Backwards 3.MD.D.8

The finished perimeter is 108 cm, so 128 minus 4 times OC equals 108. That means 4 times OC equals 20, so OC equals 5.

128 - 4 \times OC = 108 \;\Rightarrow\; 4 \times OC = 20 \;\Rightarrow\; OC = 5 \text{ cm}

O sits 5 cm above C.

Answer: 5 cm

4 · Reviewdoes it hold up?

Put 5 back in: 128 - 4 x 5 = 128 - 20 = 108 cm, the perimeter given.

Another way: A quick sense check on the size: OC must be less than half of DC, which is 12 / 2 = 6 cm, because O is nearer C than D. 5 cm fits.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Writing and unpicking the perimeter of the completed figure.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Reading the half turn about O and finding what it hides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — The perimeter of one rectangle from its two side lengths.

💡Takeaway. When two shapes are joined, whatever they share stops being on the outside -- and it stops counting once for each shape.

A half turn hides the overlap twice over · 12 practice problems

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