Lesson 14
Side Length Quotients in Similar Triangles
Let’s find missing side lengths in triangles.
Problem 1
These two triangles are similar. What are \(a\) and \(b\)? Note: the two figures are not drawn to scale.
![Two triangles. First with sides 10, 15, b. Sides with length 10 and 15 form an obtuse angle. Second with sides 4, a, 9. Sides with length 4 and a, form an obtuse angle.](https://cms-im.s3.amazonaws.com/MKnnUUhasB29CG7XttXjGzCK?response-content-disposition=inline%3B%20filename%3D%228-8.2.B9.newPP.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B9.newPP.Image.02.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=20240727T024152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ede65a5cdbf027205d980a1f219b57159e34a8139b139ddcca24174b15cb042c)
Problem 2
Here is triangle \(ABC\). Triangle \(XYZ\) is similar to \(ABC\) with scale factor \(\frac 1 4\).
![Triangle A, B C. Side A, B length 4, side B C length 7, side C A, length 5.](https://cms-im.s3.amazonaws.com/1e2mx71W2GSXx9HMjBZdAw2D?response-content-disposition=inline%3B%20filename%3D%228-8.2.B4.Image.02.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B4.Image.02.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=20240727T024152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b7ae11cec863fba4926f1f2979d4a56a806e46b5e946ce97a9c49ec2484be4b4)
- Draw what triangle \(XYZ\) might look like.
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How do the angle measures of triangle \(XYZ\) compare to triangle \(ABC\)? Explain how you know.
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What are the side lengths of triangle \(XYZ\)?
- For triangle \(XYZ\), calculate (long side) \(\div\) (medium side), and compare to triangle \(ABC\).
Problem 3
The two triangles shown are similar. Find the value of \(\frac d c\).
![Two right triangles with each hypotenuse on the same line. First has horizontal side length 7 point 5, vertical side length 9. Second has horizontal side length d and vertical side length c.](https://cms-im.s3.amazonaws.com/4zCitsPZYeZHq7pCtqazhTxj?response-content-disposition=inline%3B%20filename%3D%228-8.2.B9.newPP.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B9.newPP.Image.01.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=20240727T024152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b1d047a087b44ec21a9234edcd9947f1a3ce15d31db78006815650dc4b044a39)
Problem 4
The diagram shows two nested triangles that share a vertex. Find a center and a scale factor for a dilation that would move the larger triangle to the smaller triangle.
![Coordinate plane, x, negative 9 to 3, y, negative 2 to 7.](https://cms-im.s3.amazonaws.com/7nQZy4PiifYzszNghrWAo8Zj?response-content-disposition=inline%3B%20filename%3D%228-8.2.A.PP.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.A.PP.Image.12.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=20240727T024152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ee69c8bbe3cb2b49ee752a0340e1a2195c7ffb9614b523e127e4e233a33cb775)
Problem 5
Which is a scaled copy of Polygon A? Identify a pair of corresponding sides and a pair of corresponding angles. Compare the areas of the scaled copies.
![Six L-shaped figures on a grid, labeled A, B, C, D, E, and F.](https://cms-im.s3.amazonaws.com/dhFDv9oFeD7mW2zKpynNKtHg?response-content-disposition=inline%3B%20filename%3D%227-7.1.A.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%277-7.1.A.PP.Image.05.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=20240727T024152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cd077cf4061a44eb56bf9ae17834c75b3c9a4ffe7d7e0b131bdad3bc83a6e940)
Problem 6
A map of Colorado says that the scale is 1 inch to 20 miles or 1 to 1,267,200. Are these two ways of reporting the scale the same? Explain your reasoning.