Lesson 8
Finding Unknown Side Lengths
Let’s find missing side lengths of right triangles.
Problem 1
Find the exact value of each variable that represents a side length in a right triangle.
![5 right triangles.](https://cms-im.s3.amazonaws.com/2sRVrp49iHzx5cdjPjgryw6k?response-content-disposition=inline%3B%20filename%3D%228-8.8.C8.PP.Image.0002.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.C8.PP.Image.0002.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=20240726T233551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=87b9cb9c07ad7da74cd97593dd6f42b0db9646cfef50f0dfea635837be5e6386)
Problem 2
A right triangle has side lengths of \(a\), \(b\), and \(c\) units. The longest side has a length of \(c\) units. Complete each equation to show three relations among \(a\), \(b\), and \(c\).
-
\(c^2=\)
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\(a^2=\)
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\(b^2=\)
Problem 3
What is the exact length of each line segment? Explain or show your reasoning. (Each grid square represents 1 square unit.)
a.
b.
c.
![A line segment labeled l on a square grid. One endpoint is 4 units directly down from the other endpoint.](https://cms-im.s3.amazonaws.com/Rde6QqyjDVGTyhrm95EAaHuf?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.19.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.19.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=20240726T233551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c7b61100fff7dacaa8b1d3b21110cceff1339fe5dcb61a4096074b728126ada5)
![A line segment slanted upward and to the right, labeled m, on a square grid. The top endpoint is 2 units up and 4 units to the right from the bottom endpoint.](https://cms-im.s3.amazonaws.com/fKb2Dp3qeMfj4MehbCjQyexJ?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.20.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.20.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=20240726T233551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b264fd1026094227d5abfeba4e2607014b44c01a6978bf6964332186683f2279)
![A line segment labeled “q” on a square grid.](https://cms-im.s3.amazonaws.com/Dwxr2fPaPzrHV76eoDoPTMHH?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.23.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.23.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=20240726T233551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b114fa13415dbddbdf587c4a5d69a826de03c8cae69818040a8b179e98fbf5d3)
Problem 4
In 2015, there were roughly \(1 \times 10^6\) high school football players and \(2 \times 10^3\) professional football players in the United States. About how many times more high school football players are there? Explain how you know.
Problem 5
Evaluate:
- \(\left(\frac{1}{2}\right)^3\)
- \(\left(\frac{1}{2}\right)^{\text-3}\)
Problem 6
Here is a scatter plot of weight vs. age for different Dobermans. The model, represented by \(y = 2.45x + 1.22\), is graphed with the scatter plot. Here, \(x\) represents age in weeks, and \(y\) represents weight in pounds.
![Scatter plot with line of best fit. Horizontal axis, age in weeks, scale 0 to 25, by 5’s. Vertical axis, weight in pounds, scale 0 to 80, by 20’s.](https://cms-im.s3.amazonaws.com/s7ZweZsToQq7DabFJAudgcYA?response-content-disposition=inline%3B%20filename%3D%228-8.6.B7.PP.doberman1trend.png%22%3B%20filename%2A%3DUTF-8%27%278-8.6.B7.PP.doberman1trend.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=20240726T233551Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9ba8ee7abb2870deec2dfeb52f7d28b43d17e916e63359ab29069fd0fa04941f)
- What does the slope mean in this situation?
- Based on this model, how heavy would you expect a newborn Doberman to be?