← Diagonals of a parallelogram bisect each other · Quadrilateral Diagonal Properties

Diagonals of a parallelogram bisect each other · 8 practice problems

4.G.A.24.MD.A.3

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Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $7\,\text{cm}$ and side $AB$ measures $5\,\text{cm}$ . The full diagonal $BD$ measures $6\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $5.4\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 7 cm, AB = 5 cm, the whole diagonal BD = 6 cm, and the half-diagonal AM = 5.4 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 7 cm and AB = 5 cm.
Full diagonal BD = 6 cm.
Half-diagonal AM = 5.4 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 6 / 2 = 3 cm.

6 \div 2 = 3 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 6 cm diagonal is 3 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 5.4 cm.

CM = AM = 5.4 \text{ cm}

Since M halves AC, the piece from M to C is the same 5.4 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 7 cm.

BC = AD = 7 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 7 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 3 + 5.4 + 7 = 15.4 cm.

3 + 5.4 + 7 = 15.4 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 15.4 cm

4 · Reviewdoes it hold up?

The three sides 3 cm, 5.4 cm, and 7 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 15.4 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 2 easy

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $5\,\text{cm}$ and side $AB$ measures $3\,\text{cm}$ . The full diagonal $BD$ measures $8\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $4.5\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 5 cm, AB = 3 cm, the whole diagonal BD = 8 cm, and the half-diagonal AM = 4.5 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 5 cm and AB = 3 cm.
Full diagonal BD = 8 cm.
Half-diagonal AM = 4.5 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 8 / 2 = 4 cm.

8 \div 2 = 4 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 8 cm diagonal is 4 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 4.5 cm.

CM = AM = 4.5 \text{ cm}

Since M halves AC, the piece from M to C is the same 4.5 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 5 cm.

BC = AD = 5 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 5 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 4 + 4.5 + 5 = 13.5 cm.

4 + 4.5 + 5 = 13.5 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 13.5 cm

4 · Reviewdoes it hold up?

The three sides 4 cm, 4.5 cm, and 5 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 13.5 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 3 medium

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $9\,\text{cm}$ and side $AB$ measures $5\,\text{cm}$ . The full diagonal $BD$ measures $8\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $6.5\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 9 cm, AB = 5 cm, the whole diagonal BD = 8 cm, and the half-diagonal AM = 6.5 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 9 cm and AB = 5 cm.
Full diagonal BD = 8 cm.
Half-diagonal AM = 6.5 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 8 / 2 = 4 cm.

8 \div 2 = 4 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 8 cm diagonal is 4 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 6.5 cm.

CM = AM = 6.5 \text{ cm}

Since M halves AC, the piece from M to C is the same 6.5 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 9 cm.

BC = AD = 9 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 9 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 4 + 6.5 + 9 = 19.5 cm.

4 + 6.5 + 9 = 19.5 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 19.5 cm

4 · Reviewdoes it hold up?

The three sides 4 cm, 6.5 cm, and 9 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 19.5 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 4 medium

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $10\,\text{cm}$ and side $AB$ measures $7\,\text{cm}$ . The full diagonal $BD$ measures $6\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $8.2\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 10 cm, AB = 7 cm, the whole diagonal BD = 6 cm, and the half-diagonal AM = 8.2 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 10 cm and AB = 7 cm.
Full diagonal BD = 6 cm.
Half-diagonal AM = 8.2 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 6 / 2 = 3 cm.

6 \div 2 = 3 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 6 cm diagonal is 3 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 8.2 cm.

CM = AM = 8.2 \text{ cm}

Since M halves AC, the piece from M to C is the same 8.2 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 10 cm.

BC = AD = 10 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 10 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 3 + 8.2 + 10 = 21.2 cm.

3 + 8.2 + 10 = 21.2 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 21.2 cm

4 · Reviewdoes it hold up?

The three sides 3 cm, 8.2 cm, and 10 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 21.2 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 5 medium

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $8\,\text{cm}$ and side $AB$ measures $6\,\text{cm}$ . The full diagonal $BD$ measures $10\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $4\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 8 cm, AB = 6 cm, the whole diagonal BD = 10 cm, and the half-diagonal AM = 4 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 8 cm and AB = 6 cm.
Full diagonal BD = 10 cm.
Half-diagonal AM = 4 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 10 / 2 = 5 cm.

10 \div 2 = 5 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 10 cm diagonal is 5 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 4 cm.

CM = AM = 4 \text{ cm}

Since M halves AC, the piece from M to C is the same 4 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 8 cm.

BC = AD = 8 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 8 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 5 + 4 + 8 = 17 cm.

5 + 4 + 8 = 17 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 17 cm

4 · Reviewdoes it hold up?

The three sides 5 cm, 4 cm, and 8 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 17 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 6 hard

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $12\,\text{cm}$ and side $AB$ measures $9\,\text{cm}$ . The full diagonal $BD$ measures $10\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $8\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 12 cm, AB = 9 cm, the whole diagonal BD = 10 cm, and the half-diagonal AM = 8 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 12 cm and AB = 9 cm.
Full diagonal BD = 10 cm.
Half-diagonal AM = 8 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 10 / 2 = 5 cm.

10 \div 2 = 5 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 10 cm diagonal is 5 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 8 cm.

CM = AM = 8 \text{ cm}

Since M halves AC, the piece from M to C is the same 8 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 12 cm.

BC = AD = 12 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 12 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 5 + 8 + 12 = 25 cm.

5 + 8 + 12 = 25 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 25 cm

4 · Reviewdoes it hold up?

The three sides 5 cm, 8 cm, and 12 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 25 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 7 medium

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $6\,\text{cm}$ and side $AB$ measures $4\,\text{cm}$ . The full diagonal $BD$ measures $12\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $7\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 6 cm, AB = 4 cm, the whole diagonal BD = 12 cm, and the half-diagonal AM = 7 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 6 cm and AB = 4 cm.
Full diagonal BD = 12 cm.
Half-diagonal AM = 7 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 12 / 2 = 6 cm.

12 \div 2 = 6 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 12 cm diagonal is 6 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 7 cm.

CM = AM = 7 \text{ cm}

Since M halves AC, the piece from M to C is the same 7 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 6 cm.

BC = AD = 6 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 6 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 6 + 7 + 6 = 19 cm.

6 + 7 + 6 = 19 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 19 cm

4 · Reviewdoes it hold up?

The three sides 6 cm, 7 cm, and 6 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 19 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Variant 8 hard

In parallelogram $ABCD$ , find the perimeter, in centimeters, of triangle $BCM$ .

[Figure] Parallelogram $ABCD$ has its top side running from $A$ (top-left) to $D$ (top-right) and its bottom side from $B$ (bottom-left) to $C$ (bottom-right). The two diagonals $AC$ and $BD$ meet at point $M$ . Side $AD$ measures $7\,\text{cm}$ and side $AB$ measures $6\,\text{cm}$ . The full diagonal $BD$ measures $14\,\text{cm}$ , and along diagonal $AC$ the segment $AM$ is labeled $9.5\,\text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

In parallelogram ABCD the two diagonals AC and BD cross at M. I know AD = 7 cm, AB = 6 cm, the whole diagonal BD = 14 cm, and the half-diagonal AM = 9.5 cm. I need the perimeter of triangle BCM (the three sides BC, CM, MB added up).

Givens

ABCD is a parallelogram with diagonals AC and BD meeting at M.
AD = 7 cm and AB = 6 cm.
Full diagonal BD = 14 cm.
Half-diagonal AM = 9.5 cm.

Unknowns

The perimeter of triangle BCM, i.e. BC + CM + MB.

Constraints

The diagonals of a parallelogram bisect each other, so M is the midpoint of both BD and AC.
Opposite sides of a parallelogram are equal, so BC = AD.

2 · Planchoose the strategy

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

Find the three sides of triangle BCM one at a time using two parallelogram facts: diagonals bisect each other (gives MB and CM) and opposite sides are equal (gives BC). Then add.

3 · Execute4 carry out the plan

1Find MB from the bisected diagonal BD

#7 Identify Subproblems 4.G.A.2

M is the midpoint of diagonal BD because a parallelogram's diagonals cut each other in half. So MB = BD / 2 = 14 / 2 = 7 cm.

14 \div 2 = 7 \text{ cm}

Diagonals of a parallelogram split each other evenly, so half of the 14 cm diagonal is 7 cm.

2Find CM from the bisected diagonal AC

#7 Identify Subproblems 4.G.A.2

M is also the midpoint of diagonal AC, so CM equals the other half AM. Therefore CM = AM = 9.5 cm.

CM = AM = 9.5 \text{ cm}

Since M halves AC, the piece from M to C is the same 9.5 cm as the piece from A to M.

3Find BC from opposite sides

#1 Draw a Diagram 4.G.A.2

In a parallelogram opposite sides are equal, and BC is opposite AD, so BC = AD = 7 cm.

BC = AD = 7 \text{ cm}

Opposite sides of a parallelogram match, so BC copies AD's 7 cm.

4Add the three sides

#7 Identify Subproblems 4.MD.A.3

Perimeter of triangle BCM = MB + CM + BC = 7 + 9.5 + 7 = 23.5 cm.

7 + 9.5 + 7 = 23.5 \text{ cm}

Adding the three side lengths gives the distance all the way around the triangle.

Answer: 23.5 cm

4 · Reviewdoes it hold up?

The three sides 7 cm, 9.5 cm, and 7 cm each obey the triangle rule (the two shorter sides sum to more than the longest), so a real triangle exists. The total 23.5 cm is a sensible perimeter in centimeters.

Another way: Draw the diagram (tool 1) and mark the two equal halves on each diagonal; reading MB, MC, and BC straight off the marked figure gives the same lengths.

Standardsmin grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using parallelogram properties: diagonals bisect each other and opposite sides are equal.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Adding the three side lengths to get the perimeter of triangle BCM.

💡Takeaway. A parallelogram's diagonals chop each other exactly in half, so half-lengths plus a matching side give the perimeter with just Grade 4 addition!

Where next?

← Back to the problem: Diagonals of a parallelogram bisect each other