Practice · Corresponding sides of congruent figures are equal · Sensim Math

Variant 1 easy answer: 46 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 9 cm and AB = 8 cm.
In triangle BED: ED = 6 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 8 cm gives BE = 8 cm. ED = 6 cm gives BC = 6 cm. CA = 9 cm gives DB = 9 cm.

BE = AB = 8,\quad BC = ED = 6,\quad DB = CA = 9

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 8 + 6 + 9 = 23 cm.

(8 + 6 + 9) + (8 + 6 + 9) = 23 + 23 = 46

The way around is 46 cm.

Answer: 46 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 23 cm -- and 2 x 23 = 46 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 2 easy answer: 56 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 12 cm and AB = 9 cm.
In triangle BED: ED = 7 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 9 cm gives BE = 9 cm. ED = 7 cm gives BC = 7 cm. CA = 12 cm gives DB = 12 cm.

BE = AB = 9,\quad BC = ED = 7,\quad DB = CA = 12

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 9 + 7 + 12 = 28 cm.

(9 + 7 + 12) + (9 + 7 + 12) = 28 + 28 = 56

The way around is 56 cm.

Answer: 56 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 28 cm -- and 2 x 28 = 56 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 3 easy answer: 56 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 13 cm and AB = 10 cm.
In triangle BED: ED = 5 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 10 cm gives BE = 10 cm. ED = 5 cm gives BC = 5 cm. CA = 13 cm gives DB = 13 cm.

BE = AB = 10,\quad BC = ED = 5,\quad DB = CA = 13

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 10 + 5 + 13 = 28 cm.

(10 + 5 + 13) + (10 + 5 + 13) = 28 + 28 = 56

The way around is 56 cm.

Answer: 56 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 28 cm -- and 2 x 28 = 56 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 4 easy answer: 64 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 15 cm and AB = 6 cm.
In triangle BED: ED = 11 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 6 cm gives BE = 6 cm. ED = 11 cm gives BC = 11 cm. CA = 15 cm gives DB = 15 cm.

BE = AB = 6,\quad BC = ED = 11,\quad DB = CA = 15

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 6 + 11 + 15 = 32 cm.

(6 + 11 + 15) + (6 + 11 + 15) = 32 + 32 = 64

The way around is 64 cm.

Answer: 64 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 32 cm -- and 2 x 32 = 64 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 5 medium answer: 74 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 16 cm and AB = 12 cm.
In triangle BED: ED = 9 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 12 cm gives BE = 12 cm. ED = 9 cm gives BC = 9 cm. CA = 16 cm gives DB = 16 cm.

BE = AB = 12,\quad BC = ED = 9,\quad DB = CA = 16

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 12 + 9 + 16 = 37 cm.

(12 + 9 + 16) + (12 + 9 + 16) = 37 + 37 = 74

The way around is 74 cm.

Answer: 74 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 37 cm -- and 2 x 37 = 74 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 6 medium answer: 78 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 17 cm and AB = 14 cm.
In triangle BED: ED = 8 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 14 cm gives BE = 14 cm. ED = 8 cm gives BC = 8 cm. CA = 17 cm gives DB = 17 cm.

BE = AB = 14,\quad BC = ED = 8,\quad DB = CA = 17

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 14 + 8 + 17 = 39 cm.

(14 + 8 + 17) + (14 + 8 + 17) = 39 + 39 = 78

The way around is 78 cm.

Answer: 78 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 39 cm -- and 2 x 39 = 78 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 7 medium answer: 88 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 19 cm and AB = 13 cm.
In triangle BED: ED = 12 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 13 cm gives BE = 13 cm. ED = 12 cm gives BC = 12 cm. CA = 19 cm gives DB = 19 cm.

BE = AB = 13,\quad BC = ED = 12,\quad DB = CA = 19

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 13 + 12 + 19 = 44 cm.

(13 + 12 + 19) + (13 + 12 + 19) = 44 + 44 = 88

The way around is 88 cm.

Answer: 88 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 44 cm -- and 2 x 44 = 88 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 8 medium answer: 92 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 20 cm and AB = 15 cm.
In triangle BED: ED = 11 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 15 cm gives BE = 15 cm. ED = 11 cm gives BC = 11 cm. CA = 20 cm gives DB = 20 cm.

BE = AB = 15,\quad BC = ED = 11,\quad DB = CA = 20

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 15 + 11 + 20 = 46 cm.

(15 + 11 + 20) + (15 + 11 + 20) = 46 + 46 = 92

The way around is 92 cm.

Answer: 92 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 46 cm -- and 2 x 46 = 92 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 9 hard answer: 98 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 21 cm and AB = 18 cm.
In triangle BED: ED = 10 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 18 cm gives BE = 18 cm. ED = 10 cm gives BC = 10 cm. CA = 21 cm gives DB = 21 cm.

BE = AB = 18,\quad BC = ED = 10,\quad DB = CA = 21

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 18 + 10 + 21 = 49 cm.

(18 + 10 + 21) + (18 + 10 + 21) = 49 + 49 = 98

The way around is 98 cm.

Answer: 98 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 49 cm -- and 2 x 49 = 98 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 10 hard answer: 116 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 25 cm and AB = 20 cm.
In triangle BED: ED = 13 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 20 cm gives BE = 20 cm. ED = 13 cm gives BC = 13 cm. CA = 25 cm gives DB = 25 cm.

BE = AB = 20,\quad BC = ED = 13,\quad DB = CA = 25

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 20 + 13 + 25 = 58 cm.

(20 + 13 + 25) + (20 + 13 + 25) = 58 + 58 = 116

The way around is 116 cm.

Answer: 116 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 58 cm -- and 2 x 58 = 116 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 11 hard answer: 114 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 26 cm and AB = 7 cm.
In triangle BED: ED = 24 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 7 cm gives BE = 7 cm. ED = 24 cm gives BC = 24 cm. CA = 26 cm gives DB = 26 cm.

BE = AB = 7,\quad BC = ED = 24,\quad DB = CA = 26

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 7 + 24 + 26 = 57 cm.

(7 + 24 + 26) + (7 + 24 + 26) = 57 + 57 = 114

The way around is 114 cm.

Answer: 114 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 57 cm -- and 2 x 57 = 114 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Variant 12 hard answer: 138 cm

Triangle $ABC$ and triangle $BED$ are congruent to each other. Find the perimeter of the whole figure, in cm.

Show solution

1 · Understandwhat's really being asked

Two congruent triangles, ABC and BED, are joined at the single point B. Three of the six sides are labelled; we need the distance around the whole figure.

Givens

In triangle ABC: AC = 30 cm and AB = 22 cm.
In triangle BED: ED = 17 cm.
The two triangles are congruent.

Unknowns

The perimeter of the two triangles together.

Constraints

The triangles meet only at the point B, so no side is hidden between them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#5 Look for a Pattern

Match the vertices in the order the naming gives, copy each known length onto its partner side, and then simply walk the outside.

3 · Execute3 carry out the plan

1Match the corresponding sides

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

Because triangle ABC is congruent to triangle BED, the vertices match in the order written: A with B, B with E, C with D. So side AB matches side BE, side BC matches side ED, and side CA matches side DB.

AB \leftrightarrow BE,\quad BC \leftrightarrow ED,\quad CA \leftrightarrow DB

The naming order is the matching.

2Copy the missing lengths from the partner sides

#5 Look for a Pattern 4.G.A.1

Corresponding sides are equal, so each known length tells me its twin. AB = 22 cm gives BE = 22 cm. ED = 17 cm gives BC = 17 cm. CA = 30 cm gives DB = 30 cm.

BE = AB = 22,\quad BC = ED = 17,\quad DB = CA = 30

All six sides are now known.

3Add up all six outside sides

#7 Identify Subproblems 3.MD.D.8

The two triangles touch only at point B, so nothing is tucked inside -- the perimeter is the sum of all six sides. Each triangle contributes 22 + 17 + 30 = 69 cm.

(22 + 17 + 30) + (22 + 17 + 30) = 69 + 69 = 138

The way around is 138 cm.

Answer: 138 cm

4 · Reviewdoes it hold up?

Congruent figures have equal perimeters, so the answer has to be exactly twice one triangle's 69 cm -- and 2 x 69 = 138 cm.

Another way: If the two triangles had been joined along a whole side instead of at a point, that side would be hidden twice and the perimeter would be smaller. Meeting at a single point hides nothing.

Standardsmin grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the six outside sides into a perimeter.
4.G.A.1 Draw and identify lines and angles, and classify shapes by their properties — Reading the naming order as the correspondence between sides.

💡Takeaway. The order the letters are written in tells you which side matches which, so one labelled side gives you two.

Corresponding sides of congruent figures are equal · 12 practice problems

Generated variants — 12

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