Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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help me answer this question please
![**Using Proportions to Solve for x and y**
The image depicts a right triangle with a smaller right triangle inside it, forming similar triangles. The goal is to solve for the unknown lengths \( x \) and \( y \).
Here's a detailed explanation:
1. Identify the triangles:
- The larger triangle is a right triangle with the legs labeled as 5 and \( x \), and the hypotenuse as the side y.
- The smaller right triangle inside the larger one has the legs labeled 3 and y, and the hypotenuse as part of the larger triangle's side.
- Note the right angle marker indicating that the triangles are similar by the Angle-Angle (AA) similarity postulate.
2. Use the proportions of the similar triangles to set up the equations.
Since the triangles are similar, the ratios of their corresponding sides will be equal.
From the similarity:
\[
\frac{3}{5} = \frac{y}{x}
\]
3. Solving the proportions:
\[
x = \frac{5}{3}y
\]
Now use the Pythagorean theorem for the smaller triangle:
\[
3^2 + y^2 = x^2 \also \frac{5}{5}x
\]
Substituting \ x = \ y } into the theorem:
\x{y}{5} = y
So we can see \( x^2 = \ 9+ y^2
Using these proportions will help to determine the values of x and y in the triangle.
This exercise illustrates the application of similar triangles and proportional reasoning in solving geometric problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdadbed77-0193-4e9d-88df-390defe35d5d%2Fc9c26a55-1b17-4162-ab28-e8b30178451a%2F5fit1fm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Proportions to Solve for x and y**
The image depicts a right triangle with a smaller right triangle inside it, forming similar triangles. The goal is to solve for the unknown lengths \( x \) and \( y \).
Here's a detailed explanation:
1. Identify the triangles:
- The larger triangle is a right triangle with the legs labeled as 5 and \( x \), and the hypotenuse as the side y.
- The smaller right triangle inside the larger one has the legs labeled 3 and y, and the hypotenuse as part of the larger triangle's side.
- Note the right angle marker indicating that the triangles are similar by the Angle-Angle (AA) similarity postulate.
2. Use the proportions of the similar triangles to set up the equations.
Since the triangles are similar, the ratios of their corresponding sides will be equal.
From the similarity:
\[
\frac{3}{5} = \frac{y}{x}
\]
3. Solving the proportions:
\[
x = \frac{5}{3}y
\]
Now use the Pythagorean theorem for the smaller triangle:
\[
3^2 + y^2 = x^2 \also \frac{5}{5}x
\]
Substituting \ x = \ y } into the theorem:
\x{y}{5} = y
So we can see \( x^2 = \ 9+ y^2
Using these proportions will help to determine the values of x and y in the triangle.
This exercise illustrates the application of similar triangles and proportional reasoning in solving geometric problems.
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