Lesson 8
Area of Triangles
Problem 1
To find the area of this right triangle, Diego and Jada used different strategies. Diego drew a line through the midpoints of the two longer sides, which decomposes the triangle into a trapezoid and a smaller triangle. He then rearranged the two shapes into a parallelogram.
![A triangle decomposed into a trapezoid and a smaller triangle, then rearranged into a parallelogram.](https://cms-im.s3.amazonaws.com/J8FJmTvn5p8DTtjznVbafeXL?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e0cef9e15ac2f603695d1b00539de9e34852137bb52358a7b8b9d5e156920a5d)
Jada made a copy of the triangle, rotated it, and lined it up against one side of the original triangle so that the two triangles make a parallelogram.
![A triangle with one side labeled 3 feet and another labeled 8 ft. To the left is the same triangle with a copy composed along the 8 feet side to create a parallelogram.](https://cms-im.s3.amazonaws.com/1Q4TkpJYtAVdPYiP2c8UMT8A?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.06.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.06.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=10ed2f4c6912056c955ede712b858e6a13974e303f7957288922e8a7bf9aaa50)
- Explain how Diego might use his parallelogram to find the area of the triangle.
- Explain how Jada might use her parallelogram to find the area of the triangle.
Solution
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Problem 2
Find the area of the triangle. Explain or show your reasoning.
a.
![triangle on a grid, base = 6 units, height = 4 units.](https://cms-im.s3.amazonaws.com/GqD83JC9VNorUeXaKTF51E2o?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_10.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_10.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b8368e5c8a69481ffa9dbbc8baa1f67108aa2c7bf41fd9b8d59cc62fd5831015)
b.
![Triangle on grid, base 4, height 3 units.](https://cms-im.s3.amazonaws.com/Y3FD4EteZ2eoiopmBaVDuF7s?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_8.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_8.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8ccdaf6ef7fee38a17db7c80f3b3df2ac69085ac92e6d487dbd5905c8cf933ca)
Solution
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Problem 3
Which of the three triangles has the greatest area? Show your reasoning. If you get stuck, try using what you know about the area of parallelograms.
![Three triangles labeled A, B, and C. Triangle A is a right triangle with a base of 5 and a height of 4. Triangle B has a base of 4 and a height of 5. Triangle C has a base of 4 and a height of 5.](https://cms-im.s3.amazonaws.com/aTmHuSfaFgCDdDMHatpMBXTH?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f13c46db66400131a8f601bad1e1d8967d88b7da5c5cf4172bc7eb33a92f2152)
Solution
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Problem 4
Draw an identical copy of each triangle such that the two copies together form a parallelogram. If you get stuck, consider using tracing paper.
![Three triangles labeled D, E, and F.](https://cms-im.s3.amazonaws.com/VE8UhwU2Ty31oAiAXpX7GNuu?response-content-disposition=inline%3B%20filename%3D%226-6.1.C1.Image.15b.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C1.Image.15b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T001102Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7b12122a1f407458b05ef28271108cc78d4390d1f694e4dca0d28c04511c9342)
Solution
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(From Unit 1, Lesson 7.)Problem 5
-
A parallelogram has a base of 3.5 units and a corresponding height of 2 units. What is its area?
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A parallelogram has a base of 3 units and an area of 1.8 square units. What is the corresponding height for that base?
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A parallelogram has an area of 20.4 square units. If the height that corresponds to a base is 4 units, what is the base?
Solution
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(From Unit 1, Lesson 6.)