Lesson 3
Scaled Relationships
Problem 1
Here is Quadrilateral \(ABCD\).
![Quadrilateral ABCD is on a grid.](https://staging-cms-im.s3.amazonaws.com/iNwd9oUU8CeHiBnW4T3iCWta?response-content-disposition=inline%3B%20filename%3D%227-7.1.PP.New.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%277-7.1.PP.New.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234724Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1d091943b041330e34239e62e89b4fc0bb4b2ada3f2a4243fdaeeaa2a8698da5)
Quadrilateral \(PQRS\) is a scaled copy of Quadrilateral \(ABCD\). Point \(P\) corresponds to \(A\), \(Q\) to \(B\), \(R\) to \(C\), and \(S\) to \(D\).
If the distance from \(P\) to \(R\) is 3 units, what is the distance from \(Q\) to \(S\)? Explain your reasoning.
Problem 2
Rectangles P, Q, R, and S are scaled copies of one another. For each pair, decide if the scale factor from one to the other is greater than 1, equal to 1, or less than 1.
![Four rectangles, labeled P, Q, R and S.](https://staging-cms-im.s3.amazonaws.com/r92sCFtwgkGK3qLKRXxxY76s?response-content-disposition=inline%3B%20filename%3D%227-7.1.A.PP.Image.22.png%22%3B%20filename%2A%3DUTF-8%27%277-7.1.A.PP.Image.22.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234724Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2c727063dda5025232042172304534a83cabf1de298c567082351e302ed6dd48)
- from P to Q
- from P to R
- from Q to S
- from Q to R
- from S to P
- from R to P
- from P to S
Problem 3
Triangle S and Triangle L are scaled copies of one another.
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What is the scale factor from S to L?
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What is the scale factor from L to S?
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Triangle M is also a scaled copy of S. The scale factor from S to M is \(\frac{3}{2}\). What is the scale factor from M to S?
![Two triangles labeled S and L on a grid. Triangle S has a horizontal base of 2 units and a height of 4 units. Triangle L has a horizontal base of 4 units and a height of 8 units.](https://staging-cms-im.s3.amazonaws.com/QzSG8Ti4pi8B684RjAQQEcZQ?response-content-disposition=inline%3B%20filename%3D%227-7.1.A.PP.Image.31.png%22%3B%20filename%2A%3DUTF-8%27%277-7.1.A.PP.Image.31.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234724Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2147c09928e0a66c16f8d067b7968735be76648d6ff9ef0fcebfa03474676a6e)
Problem 4
Are two squares with the same side lengths scaled copies of one another? Explain your reasoning.
Problem 5
Quadrilateral A has side lengths 2, 3, 5, and 6. Quadrilateral B has side lengths 4, 5, 8, and 10. Could one of the quadrilaterals be a scaled copy of the other? Explain.
Problem 6
The line has been partitioned into three angles.
![A straight line with two rays coming out of a single point.](https://staging-cms-im.s3.amazonaws.com/dc5os7EpTNp9H4DanFhYwXFt?response-content-disposition=inline%3B%20filename%3D%228-8.1.D.PP.Image.04.5.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.D.PP.Image.04.5.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234724Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3e85a0bf7cc63225636f0ffb9a381dd00bb7c2be7ebaded6cdf2c980e5adcf88)
Is there a triangle with these three angle measures? Explain.