Lesson 6
A Proof of the Pythagorean Theorem
Let’s prove the Pythagorean Theorem.
Problem 1
- Find the lengths of the unlabeled sides.
- One segment is \(n\) units long and the other is \(p\) units long. Find the value of \(n\) and \(p\). (Each small grid square is 1 square unit.)
Problem 2
Use the areas of the two identical squares to explain why \(5^2+12^2=13^2\) without doing any calculations.
![2 decomposed squares.](https://cms-im.s3.amazonaws.com/gh2AKA7gTKJjfNmFQzDih4Gs?response-content-disposition=inline%3B%20filename%3D%228-8.8.C7.PP.Image.0001.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.C7.PP.Image.0001.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T034416Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a3e1227bf109d046e00fb1d978ee93fbbb3cfb1794763619ec40cd19fe7d7fbd)
Problem 3
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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T034416Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=af153397a2ef23dfe4245cfb684bf0e1ce04814104f190bec92050c0da78d170)
Problem 4
Write each expression as a single power of 10.
- \(10^5 \boldcdot 10^0\)
- \(\frac{10^9}{10^0}\)
Problem 5
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%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T034416Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=076388dd74047b3751b3317a082e8a4b8401182cbe2d1604243650d70e44c718)
- What does the slope mean in this situation?
- Based on this model, how heavy would you expect a newborn Doberman to be?