Lesson 10
Bases and Heights of Triangles
Let’s use different base-height pairs to find the area of a triangle.
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=14edfa366d09d289ac1b3256444c3e063afd124f871e1e1af1bb647ee6c358b1)
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=016dcf341a8f55b6f3ffff2a0b772b8025c927f690b9d682d07d3ba6f85b434b)
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3410e62123489f385be0f994735d52b64d410d6b57fdd80fd551135631e62a4e)
If you get stuck, carefully consider which side of the triangle to use as the base.
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9090da8ac7c17168e1a22015e88a37dc431f69e52004e4dc9212386e6281bf77)
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2fc95d118cee23ce17ba8678f2166435fed947d1f394b54e52263a504180515b)
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=20240727T025157Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=afa680e2692b1fd9c0edec9197c1f8ff9ccbf69beb9d9a902502c7a9f1ceb175)