Lesson 9
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=20240727T025930Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=bb80e8a90dd80ff1db5d4ac364b3b4fbdad3beeebe2f659ea9f0d537a28da0e9)
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=20240727T025930Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a0d71050c3f05b2c1e5ad2ce9391bc3c4663e6a0a19cd680945dfbdb2db9f1c9)
- 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=20240727T025930Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=702ae860063f056f36126ab4b8bed7a230e830130a3b3516d2670c90d0a1de83)
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=20240727T025930Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=45ffb6d386d561dab6b00c98b8391701654d25c5cfa32702afe86e29ba4adadc)