Lesson 10
Bases and Heights of Triangles
Problem 1
For each triangle, a base is labeled \(b\). Draw a line segment that shows its corresponding height. Use an index card to help you draw a straight line.
![3 triangles with 1 side on each labeled b.](https://cms-im.s3.amazonaws.com/e7XsZDTn9zvbgdYwBN6WFXx5?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_27.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_27.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=aaef33cb73ab19ccd0b2affcefc39ba523aec8ce641e80db6c29913c0ca70d93)
Solution
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Problem 2
Select all triangles that have an area of 8 square units. Explain how you know.
![5 triangles on grid labeled A, B, C, D, E.](https://cms-im.s3.amazonaws.com/XZJAmPFrvQ59CDRFsg1U4Xfx?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_29.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_29.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4a4278055dcc41591f9314df8da2a3ff2cf2c58f39bdb8c32e1c81a6e2982067)
Solution
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Problem 3
Find the area of the triangle. Show your reasoning.
![Triangle on grid, base = 6, height = 4](https://cms-im.s3.amazonaws.com/UUQo1fWRMutt4tJgvKxPbUbr?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP_Image_30.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP_Image_30.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c7b79d055999de5283babdeeebf75cec0df37a504556a6de703c20ded0c02959)
If you get stuck, carefully consider which side of the triangle to use as the base.
Solution
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Problem 4
Can side \(d\) be the base for this triangle? If so, which length would be the corresponding height? If not, explain why not.
![A triangle with sides labeled d, e, and f. The angle opposite side D is a right angle. A segment labeled g is perpendicular to side d and extends to the opposite vertex.](https://cms-im.s3.amazonaws.com/prUqTf2JuReKCiXBRShyzV6e?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.10.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=01ca2d0924ff1543ccbe59096d47d00c300ab0d8a91b93d1e02ed1be642c386c)
Solution
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Problem 5
Find the area of this shape. Show your reasoning.
![A shape with six sides. There are two vertical sides measuring five units, two angled sides that fall 2 units over 4 units and two sides that fall 2 units over 2 units.](https://cms-im.s3.amazonaws.com/2YXc9UpF9MyRVaLHULxCRLAZ?response-content-disposition=inline%3B%20filename%3D%226-6.1.A.PP.New.Image.18a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A.PP.New.Image.18a.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=34ab380da6f13f78c2bfb60afc0c920ef8d79eb48fdfa7a5f595d2cc2de0999c)
Solution
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(From Unit 1, Lesson 3.)Problem 6
On the grid, sketch two different parallelograms that have equal area. Label a base and height of each and explain how you know the areas are the same.
![Image of a grid.](https://cms-im.s3.amazonaws.com/vupNoA5g96AkAtsSrCipKyG6?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3.Image.00.Blank-Grid.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3.Image.00.Blank-Grid.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=20240727T010551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c2e920b3260cffe10bf677df3308bc295414ad6b075996b993356deeb8a88c0e)
Solution
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(From Unit 1, Lesson 6.)