Lesson 9
Side Length Quotients in Similar 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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T150448Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e7503940baf1b6a3cf820bc2a07b9b6d90bae51229524138ee6df647e0bba7b2)
Solution
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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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T150448Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c888b099e256d26a08ecda37715a71d0c8bbfe37aaf0d1405cc5a96224c2ce6b)
- 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 \(ABC\).
Solution
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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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T150448Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=64fbc1e3bdfa4cc813bbaea3c455c65343dd57463b721ef3a809385ccf8d5674)
Solution
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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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T150448Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=81111fe25a15f53db6841062ac43ab4e45df21603f3e34708e6149558f3b41fc)
Solution
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(From Unit 2, Lesson 5.)