Practice · The right-angle mark makes the height a known number · Sensim Math

Variant 1 easy

In the figure at the right, triangle DEC has $\frac{1}{2}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 12.8 cm and 15 cm and is 8 cm tall. Triangle DEC is one 2th of it, and we want EC.

Givens

AD is 12.8 cm and BC is 15 cm.
The trapezoid is 8 cm tall.
Triangle DEC has one 2th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(12.8 + 15) \div 2 \times 8 = 111.2

The whole shape is 111.2 cm2.

2Take one 2th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

111.2 \div 2 = 55.6

Triangle DEC is 55.6 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 8 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 8

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

55.6 \times 2 \div 8 = 13.9

EC is 13.9 cm.

Answer: 13.9 cm

4 · Reviewdoes it hold up?

EC is 13.9 cm out of BC's 15 cm, so the triangle really does sit inside the trapezoid, and its area 55.6 is one 2th of 111.2.

Another way: Writing it as one equation, $\text{EC} \times 8 \div 2 = 111.2 \div 2$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 2 easy

In the figure at the right, triangle DEC has $\frac{1}{8}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 6.4 cm and 14.4 cm and is 17 cm tall. Triangle DEC is one 8th of it, and we want EC.

Givens

AD is 6.4 cm and BC is 14.4 cm.
The trapezoid is 17 cm tall.
Triangle DEC has one 8th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(6.4 + 14.4) \div 2 \times 17 = 176.8

The whole shape is 176.8 cm2.

2Take one 8th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

176.8 \div 8 = 22.1

Triangle DEC is 22.1 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 17 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 17

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

22.1 \times 2 \div 17 = 2.6

EC is 2.6 cm.

Answer: 2.6 cm

4 · Reviewdoes it hold up?

EC is 2.6 cm out of BC's 14.4 cm, so the triangle really does sit inside the trapezoid, and its area 22.1 is one 8th of 176.8.

Another way: Writing it as one equation, $\text{EC} \times 17 \div 2 = 176.8 \div 8$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 3 easy

In the figure at the right, triangle DEC has $\frac{1}{5}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 8.4 cm and 13 cm and is 17 cm tall. Triangle DEC is one 5th of it, and we want EC.

Givens

AD is 8.4 cm and BC is 13 cm.
The trapezoid is 17 cm tall.
Triangle DEC has one 5th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(8.4 + 13) \div 2 \times 17 = 181.9

The whole shape is 181.9 cm2.

2Take one 5th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

181.9 \div 5 = 36.38

Triangle DEC is 36.38 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 17 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 17

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

36.38 \times 2 \div 17 = 4.28

EC is 4.28 cm.

Answer: 4.28 cm

4 · Reviewdoes it hold up?

EC is 4.28 cm out of BC's 13 cm, so the triangle really does sit inside the trapezoid, and its area 36.38 is one 5th of 181.9.

Another way: Writing it as one equation, $\text{EC} \times 17 \div 2 = 181.9 \div 5$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 4 easy

In the figure at the right, triangle DEC has $\frac{1}{4}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 12.4 cm and 18 cm and is 8 cm tall. Triangle DEC is one 4th of it, and we want EC.

Givens

AD is 12.4 cm and BC is 18 cm.
The trapezoid is 8 cm tall.
Triangle DEC has one 4th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(12.4 + 18) \div 2 \times 8 = 121.6

The whole shape is 121.6 cm2.

2Take one 4th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

121.6 \div 4 = 30.4

Triangle DEC is 30.4 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 8 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 8

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

30.4 \times 2 \div 8 = 7.6

EC is 7.6 cm.

Answer: 7.6 cm

4 · Reviewdoes it hold up?

EC is 7.6 cm out of BC's 18 cm, so the triangle really does sit inside the trapezoid, and its area 30.4 is one 4th of 121.6.

Another way: Writing it as one equation, $\text{EC} \times 8 \div 2 = 121.6 \div 4$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 5 medium

In the figure at the right, triangle DEC has $\frac{1}{5}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 11.8 cm and 18.6 cm and is 9 cm tall. Triangle DEC is one 5th of it, and we want EC.

Givens

AD is 11.8 cm and BC is 18.6 cm.
The trapezoid is 9 cm tall.
Triangle DEC has one 5th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(11.8 + 18.6) \div 2 \times 9 = 136.8

The whole shape is 136.8 cm2.

2Take one 5th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

136.8 \div 5 = 27.36

Triangle DEC is 27.36 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 9 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 9

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

27.36 \times 2 \div 9 = 6.08

EC is 6.08 cm.

Answer: 6.08 cm

4 · Reviewdoes it hold up?

EC is 6.08 cm out of BC's 18.6 cm, so the triangle really does sit inside the trapezoid, and its area 27.36 is one 5th of 136.8.

Another way: Writing it as one equation, $\text{EC} \times 9 \div 2 = 136.8 \div 5$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 6 medium

In the figure at the right, triangle DEC has $\frac{1}{2}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 13.8 cm and 16.8 cm and is 20 cm tall. Triangle DEC is one 2th of it, and we want EC.

Givens

AD is 13.8 cm and BC is 16.8 cm.
The trapezoid is 20 cm tall.
Triangle DEC has one 2th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(13.8 + 16.8) \div 2 \times 20 = 306

The whole shape is 306 cm2.

2Take one 2th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

306 \div 2 = 153

Triangle DEC is 153 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 20 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 20

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

153 \times 2 \div 20 = 15.3

EC is 15.3 cm.

Answer: 15.3 cm

4 · Reviewdoes it hold up?

EC is 15.3 cm out of BC's 16.8 cm, so the triangle really does sit inside the trapezoid, and its area 153 is one 2th of 306.

Another way: Writing it as one equation, $\text{EC} \times 20 \div 2 = 306 \div 2$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 7 medium

In the figure at the right, triangle DEC has $\frac{1}{15}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 13.6 cm and 17 cm and is 20 cm tall. Triangle DEC is one 15th of it, and we want EC.

Givens

AD is 13.6 cm and BC is 17 cm.
The trapezoid is 20 cm tall.
Triangle DEC has one 15th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(13.6 + 17) \div 2 \times 20 = 306

The whole shape is 306 cm2.

2Take one 15th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

306 \div 15 = 20.4

Triangle DEC is 20.4 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 20 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 20

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

20.4 \times 2 \div 20 = 2.04

EC is 2.04 cm.

Answer: 2.04 cm

4 · Reviewdoes it hold up?

EC is 2.04 cm out of BC's 17 cm, so the triangle really does sit inside the trapezoid, and its area 20.4 is one 15th of 306.

Another way: Writing it as one equation, $\text{EC} \times 20 \div 2 = 306 \div 15$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 8 medium

In the figure at the right, triangle DEC has $\frac{1}{20}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 16.2 cm and 21 cm and is 6 cm tall. Triangle DEC is one 20th of it, and we want EC.

Givens

AD is 16.2 cm and BC is 21 cm.
The trapezoid is 6 cm tall.
Triangle DEC has one 20th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(16.2 + 21) \div 2 \times 6 = 111.6

The whole shape is 111.6 cm2.

2Take one 20th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

111.6 \div 20 = 5.58

Triangle DEC is 5.58 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 6 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 6

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

5.58 \times 2 \div 6 = 1.86

EC is 1.86 cm.

Answer: 1.86 cm

4 · Reviewdoes it hold up?

EC is 1.86 cm out of BC's 21 cm, so the triangle really does sit inside the trapezoid, and its area 5.58 is one 20th of 111.6.

Another way: Writing it as one equation, $\text{EC} \times 6 \div 2 = 111.6 \div 20$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 9 hard

In the figure at the right, triangle DEC has $\frac{1}{5}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 16 cm and 21.6 cm and is 14 cm tall. Triangle DEC is one 5th of it, and we want EC.

Givens

AD is 16 cm and BC is 21.6 cm.
The trapezoid is 14 cm tall.
Triangle DEC has one 5th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(16 + 21.6) \div 2 \times 14 = 263.2

The whole shape is 263.2 cm2.

2Take one 5th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

263.2 \div 5 = 52.64

Triangle DEC is 52.64 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 14 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 14

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

52.64 \times 2 \div 14 = 7.52

EC is 7.52 cm.

Answer: 7.52 cm

4 · Reviewdoes it hold up?

EC is 7.52 cm out of BC's 21.6 cm, so the triangle really does sit inside the trapezoid, and its area 52.64 is one 5th of 263.2.

Another way: Writing it as one equation, $\text{EC} \times 14 \div 2 = 263.2 \div 5$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 10 hard

In the figure at the right, triangle DEC has $\frac{1}{5}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 15.4 cm and 20.2 cm and is 22 cm tall. Triangle DEC is one 5th of it, and we want EC.

Givens

AD is 15.4 cm and BC is 20.2 cm.
The trapezoid is 22 cm tall.
Triangle DEC has one 5th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(15.4 + 20.2) \div 2 \times 22 = 391.6

The whole shape is 391.6 cm2.

2Take one 5th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

391.6 \div 5 = 78.32

Triangle DEC is 78.32 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 22 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 22

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

78.32 \times 2 \div 22 = 7.12

EC is 7.12 cm.

Answer: 7.12 cm

4 · Reviewdoes it hold up?

EC is 7.12 cm out of BC's 20.2 cm, so the triangle really does sit inside the trapezoid, and its area 78.32 is one 5th of 391.6.

Another way: Writing it as one equation, $\text{EC} \times 22 \div 2 = 391.6 \div 5$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 11 hard

In the figure at the right, triangle DEC has $\frac{1}{2}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 14 cm and 22.6 cm and is 16 cm tall. Triangle DEC is one 2th of it, and we want EC.

Givens

AD is 14 cm and BC is 22.6 cm.
The trapezoid is 16 cm tall.
Triangle DEC has one 2th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(14 + 22.6) \div 2 \times 16 = 292.8

The whole shape is 292.8 cm2.

2Take one 2th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

292.8 \div 2 = 146.4

Triangle DEC is 146.4 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 16 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 16

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

146.4 \times 2 \div 16 = 18.3

EC is 18.3 cm.

Answer: 18.3 cm

4 · Reviewdoes it hold up?

EC is 18.3 cm out of BC's 22.6 cm, so the triangle really does sit inside the trapezoid, and its area 146.4 is one 2th of 292.8.

Another way: Writing it as one equation, $\text{EC} \times 16 \div 2 = 292.8 \div 2$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

Variant 12 hard

In the figure at the right, triangle DEC has $\frac{1}{5}$ the area of trapezoid ABCD. How long is segment EC, in $\text{cm}$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A trapezoid has parallel sides 8.2 cm and 11.8 cm and is 23 cm tall. Triangle DEC is one 5th of it, and we want EC.

Givens

AD is 8.2 cm and BC is 11.8 cm.
The trapezoid is 23 cm tall.
Triangle DEC has one 5th of the trapezoid's area.
DE meets BC at a right angle.

Unknowns

The length of EC.

Constraints

E lies on BC, so EC is part of the bottom side.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram#8 Analyze the Units

The right angle at E is the key: it makes DE the trapezoid's own height, so the triangle's area formula has just one unknown left. Find the trapezoid's area, take a share of it, then undo the formula.

3 · Execute4 carry out the plan

1Find the trapezoid's area

#7 Identify Subproblems 6.G.A.1

Add the parallel sides, halve, and multiply by the height.

(8.2 + 11.8) \div 2 \times 23 = 230

The whole shape is 230 cm2.

2Take one 5th of it

#8 Analyze the Units 6.NS.B.3

That is the triangle's area.

230 \div 5 = 46

Triangle DEC is 46 cm2.

3Read the triangle's height off the right angle

#1 Draw a Diagram 6.G.A.1

DE is perpendicular to BC, so DE is the same 23 cm as the trapezoid's height. Without that mark it would be a second unknown.

\text{DE} = 23

One unknown left, not two.

4Undo the triangle's area formula

#8 Analyze the Units 6.NS.B.3

Double the area, then divide by the height.

46 \times 2 \div 23 = 4

EC is 4 cm.

Answer: 4 cm

4 · Reviewdoes it hold up?

EC is 4 cm out of BC's 11.8 cm, so the triangle really does sit inside the trapezoid, and its area 46 is one 5th of 230.

Another way: Writing it as one equation, $\text{EC} \times 23 \div 2 = 230 \div 5$ , is the same reasoning without the named intermediate values.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Using the trapezoid and triangle area formulas.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the share and then the length.

💡Takeaway. A right-angle mark is not decoration. It tells you a length you would otherwise have to find.

The right-angle mark makes the height a known number · 12 practice problems

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