### Right-Triangle Calculation Explanation In this educational exercise, we have a composite right triangle problem. There is a larger right triangle containing a smaller right triangle within it. The right triangle on the bottom left corner has sides of lengths 6 and 8, with the hypotenuse labeled as \( a \). The larger triangle shares one side with this smaller triangle and has its hypotenuse labeled as 10. The base of the larger triangle is labeled as \( b \). #### Steps to Calculate \( a \) and \( b \): 1. **Calculate Side \( a \) (Hypotenuse of the Smaller Triangle):** We can use the Pythagorean theorem to find the hypotenuse \( a \). \[ a^2 = 6^2 + 8^2 = 36 + 64 = 100 \] \[ a = \sqrt{100} = 10 \] 2. **Calculate Base \( b \) (Base of the Larger Triangle):** Since the hypotenuse of the larger triangle is 10, which is equal to the hypotenuse of the smaller triangle (and actually the base and height are the legs of the right triangle), the base \( b \) remains as given, \( b = 16 \). Therefore: \[ a = 10 \] \[ b = 16 \] You can fill in the slots provided in the diagram: \[ a = 10 \] \[ b = 16 \] This demonstrates basic usage of the Pythagorean theorem in solving problems related to right-angled triangles.
### Right-Triangle Calculation Explanation In this educational exercise, we have a composite right triangle problem. There is a larger right triangle containing a smaller right triangle within it. The right triangle on the bottom left corner has sides of lengths 6 and 8, with the hypotenuse labeled as \( a \). The larger triangle shares one side with this smaller triangle and has its hypotenuse labeled as 10. The base of the larger triangle is labeled as \( b \). #### Steps to Calculate \( a \) and \( b \): 1. **Calculate Side \( a \) (Hypotenuse of the Smaller Triangle):** We can use the Pythagorean theorem to find the hypotenuse \( a \). \[ a^2 = 6^2 + 8^2 = 36 + 64 = 100 \] \[ a = \sqrt{100} = 10 \] 2. **Calculate Base \( b \) (Base of the Larger Triangle):** Since the hypotenuse of the larger triangle is 10, which is equal to the hypotenuse of the smaller triangle (and actually the base and height are the legs of the right triangle), the base \( b \) remains as given, \( b = 16 \). Therefore: \[ a = 10 \] \[ b = 16 \] You can fill in the slots provided in the diagram: \[ a = 10 \] \[ b = 16 \] This demonstrates basic usage of the Pythagorean theorem in solving problems related to right-angled triangles.
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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![### Right-Triangle Calculation Explanation
In this educational exercise, we have a composite right triangle problem.
There is a larger right triangle containing a smaller right triangle within it. The right triangle on the bottom left corner has sides of lengths 6 and 8, with the hypotenuse labeled as \( a \). The larger triangle shares one side with this smaller triangle and has its hypotenuse labeled as 10. The base of the larger triangle is labeled as \( b \).
#### Steps to Calculate \( a \) and \( b \):
1. **Calculate Side \( a \) (Hypotenuse of the Smaller Triangle):**
We can use the Pythagorean theorem to find the hypotenuse \( a \).
\[
a^2 = 6^2 + 8^2 = 36 + 64 = 100
\]
\[
a = \sqrt{100} = 10
\]
2. **Calculate Base \( b \) (Base of the Larger Triangle):**
Since the hypotenuse of the larger triangle is 10, which is equal to the hypotenuse of the smaller triangle (and actually the base and height are the legs of the right triangle), the base \( b \) remains as given, \( b = 16 \).
Therefore:
\[
a = 10
\]
\[
b = 16
\]
You can fill in the slots provided in the diagram:
\[ a = 10 \]
\[ b = 16 \]
This demonstrates basic usage of the Pythagorean theorem in solving problems related to right-angled triangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31f394f2-9cc9-4c1e-bb28-d49c0bf01e44%2F5053cb22-cd1a-4906-9787-e73fde2cd290%2F2o0ctb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Right-Triangle Calculation Explanation
In this educational exercise, we have a composite right triangle problem.
There is a larger right triangle containing a smaller right triangle within it. The right triangle on the bottom left corner has sides of lengths 6 and 8, with the hypotenuse labeled as \( a \). The larger triangle shares one side with this smaller triangle and has its hypotenuse labeled as 10. The base of the larger triangle is labeled as \( b \).
#### Steps to Calculate \( a \) and \( b \):
1. **Calculate Side \( a \) (Hypotenuse of the Smaller Triangle):**
We can use the Pythagorean theorem to find the hypotenuse \( a \).
\[
a^2 = 6^2 + 8^2 = 36 + 64 = 100
\]
\[
a = \sqrt{100} = 10
\]
2. **Calculate Base \( b \) (Base of the Larger Triangle):**
Since the hypotenuse of the larger triangle is 10, which is equal to the hypotenuse of the smaller triangle (and actually the base and height are the legs of the right triangle), the base \( b \) remains as given, \( b = 16 \).
Therefore:
\[
a = 10
\]
\[
b = 16
\]
You can fill in the slots provided in the diagram:
\[ a = 10 \]
\[ b = 16 \]
This demonstrates basic usage of the Pythagorean theorem in solving problems related to right-angled triangles.
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