Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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![**Problem:**
Solve for the lengths of the missing sides in the triangle. Leave your answer in radical form. Show your work and explain the steps you used to solve.
**Diagram Explanation:**
The diagram depicts a right triangle with the following details:
- One angle is 30°.
- The other angle is 60°.
- The side opposite the 60° angle (hypotenuse) is labeled as 18.
- The side opposite the 30° angle is labeled as \( b \).
- The side adjacent to the 30° angle (and opposite to the 60° angle) is labeled as \( a \).
**Solution:**
In a 30°-60°-90° triangle, the ratios of the sides are 1 (shortest side): \(\sqrt{3}\) (longest side): 2 (hypotenuse).
1. **Find Side \( a \):**
Since the hypotenuse \( c \) is 18, side \( a \) (the longer leg) is \(\frac{\sqrt{3}}{2} \times c\).
\[
a = \frac{\sqrt{3}}{2} \times 18 = 9\sqrt{3}
\]
2. **Find Side \( b \):**
Since \( b \) (the shorter leg) is half of the hypotenuse:
\[
b = \frac{1}{2} \times 18 = 9
\]
**Final Answer:**
- \( a = 9\sqrt{3} \)
- \( b = 9 \)
The values have been left in radical form as requested.](https://content.bartleby.com/qna-images/question/1d7ffa32-c081-4b73-abc9-a0664169f2ac/3803c8e3-ba6a-419a-b836-945219c83dd0/pxuvxte_processed.jpeg)
Transcribed Image Text:**Problem:**
Solve for the lengths of the missing sides in the triangle. Leave your answer in radical form. Show your work and explain the steps you used to solve.
**Diagram Explanation:**
The diagram depicts a right triangle with the following details:
- One angle is 30°.
- The other angle is 60°.
- The side opposite the 60° angle (hypotenuse) is labeled as 18.
- The side opposite the 30° angle is labeled as \( b \).
- The side adjacent to the 30° angle (and opposite to the 60° angle) is labeled as \( a \).
**Solution:**
In a 30°-60°-90° triangle, the ratios of the sides are 1 (shortest side): \(\sqrt{3}\) (longest side): 2 (hypotenuse).
1. **Find Side \( a \):**
Since the hypotenuse \( c \) is 18, side \( a \) (the longer leg) is \(\frac{\sqrt{3}}{2} \times c\).
\[
a = \frac{\sqrt{3}}{2} \times 18 = 9\sqrt{3}
\]
2. **Find Side \( b \):**
Since \( b \) (the shorter leg) is half of the hypotenuse:
\[
b = \frac{1}{2} \times 18 = 9
\]
**Final Answer:**
- \( a = 9\sqrt{3} \)
- \( b = 9 \)
The values have been left in radical form as requested.
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