Lesson 7
From Parallelograms to Triangles
Problem 1
To decompose a quadrilateral into two identical shapes, Clare drew a dashed line as shown in the diagram.
![Trapezoid base 1, 6 units, base 2, 2 units, height 4 units.](https://cms-im.s3.amazonaws.com/KzBGN8Px8at5Lvees9j5RBBX?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_5.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_5.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232907Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=def6b3e12ee8b967d640d1a011d57566129d12f2ea2f6227f550fad3a45f1b9e)
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She said the that two resulting shapes have the same area. Do you agree? Explain your reasoning.
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Did Clare partition the figure into two identical shapes? Explain your reasoning.
Solution
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Problem 2
Triangle R is a right triangle. Can we use two copies of Triangle R to compose a parallelogram that is not a square?
![2 identical triangles labeled R](https://cms-im.s3.amazonaws.com/FpYzLvDFcV8sAVwXZGPrraZp?response-content-disposition=inline%3B%20filename%3D%226-6.1.C1.Image.09.R.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C1.Image.09.R.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232907Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d7178e662fc9aeb3a54771f80a048dd7249e52304679ed882d1fc3131714ef04)
If so, explain how or sketch a solution. If not, explain why not.
Solution
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Problem 3
Two copies of this triangle are used to compose a parallelogram. Which parallelogram cannot be a result of the composition? If you get stuck, consider using tracing paper.
![A three sided figure in a grid. Vertices located at two units right, 1 unit up; 7 units right, 2 units up; and 1 unit right, 3 units up.](https://cms-im.s3.amazonaws.com/VmjzaFPtL3B9MDRKts6bcxN4?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232907Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=64ba366d724ac314f46ac79169aaf8f8fd03b7be6d94c7b40a40296a7f6f0432)
![Four parallelograms labeled A, B, C, and D.](https://cms-im.s3.amazonaws.com/Gt95bSgxHEEJGwNXNBLCM9o9?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.02a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.02a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232907Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8944081ae601a0741ff8327e3372ceccb2ac317f27fa4ddd1fe5bda5e1cf7cd5)
A
B
C
D
Solution
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Problem 4
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On the grid, draw at least three different quadrilaterals that can each be decomposed into two identical triangles with a single cut (show the cut line). One or more of the quadrilaterals should have non-right angles.
- Identify the type of each quadrilateral.
Solution
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Problem 5
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A parallelogram has a base of 9 units and a corresponding height of \(\frac23\) units. What is its area?
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A parallelogram has a base of 9 units and an area of 12 square units. What is the corresponding height for that base?
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A parallelogram has an area of 7 square units. If the height that corresponds to a base is \(\frac14\) unit, what is the base?
Solution
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(From Unit 1, Lesson 6.)Problem 6
Select all the segments that could represent the height if side \(n\) is the base.
![A parallelogram with a bottom side labeled m and a right side labeled n. Dashed lines e, f, j, and k are drawn perpendicular to side m, and dashed lines g and h are drawn perpendicular to side n.](https://cms-im.s3.amazonaws.com/tFqfcBp8q4QSRV2SEENK4hVi?response-content-disposition=inline%3B%20filename%3D%226-6.1.B.PP.New.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.B.PP.New.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T232907Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2c4672bd90f657d5cd94dda689ce4713433b86a682f872b17026f593c54d1158)
\(e\)
\(f\)
\(g\)
\(h\)
\(m\)
\(n\)
\(j\)
\(k\)
Solution
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(From Unit 1, Lesson 5.)