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
Related questions
Concept explainers
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
Find the value of the missing angle X. Round your answer to the nearest tenth.
![### Right Triangle Problem
This image shows a right triangle, where one of the angles is 90 degrees (a right angle). The triangle has the following given side lengths:
- The length of one of the legs is 10 units.
- The length of the other leg is 24 units.
We need to find the length of the hypotenuse, which is marked as "X". The hypotenuse is the side opposite to the right angle and is the longest side of a right triangle.
#### Explanation and Steps:
1. **Identify the right triangle**: Recognize that one of the angles is a right angle. This fact allows us to use the Pythagorean theorem to find the unknown side length.
2. **Pythagorean Theorem**: The theorem states that in any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is:
\[
c^2 = a^2 + b^2
\]
3. **Plugging in the values**:
Given,
\[
a = 10 \text{ units}
\]
\[
b = 24 \text{ units}
\]
Therefore, the equation to find X (hypotenuse) will be:
\[
X^2 = 10^2 + 24^2
\]
\[
X^2 = 100 + 576
\]
\[
X^2 = 676
\]
4. **Solve for X**:
\[
X = \sqrt{676}
\]
\[
X = 26 \text{ units}
\]
Thus, the length of the hypotenuse (X) is 26 units.
### Summary
- Identified given side lengths of a right triangle (legs = 10 units and 24 units).
- Used the Pythagorean theorem to solve for the length of the hypotenuse.
- Found the hypotenuse length to be 26 units.
This problem illustrates the use of the Pythagorean theorem in solving for an unknown side of a right triangle, a fundamental concept in geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff66317fc-ed86-4f22-9308-86904c21cc6d%2F93470e88-374a-454d-be9a-81dcb43a2be2%2Fub6rez_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Right Triangle Problem
This image shows a right triangle, where one of the angles is 90 degrees (a right angle). The triangle has the following given side lengths:
- The length of one of the legs is 10 units.
- The length of the other leg is 24 units.
We need to find the length of the hypotenuse, which is marked as "X". The hypotenuse is the side opposite to the right angle and is the longest side of a right triangle.
#### Explanation and Steps:
1. **Identify the right triangle**: Recognize that one of the angles is a right angle. This fact allows us to use the Pythagorean theorem to find the unknown side length.
2. **Pythagorean Theorem**: The theorem states that in any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is:
\[
c^2 = a^2 + b^2
\]
3. **Plugging in the values**:
Given,
\[
a = 10 \text{ units}
\]
\[
b = 24 \text{ units}
\]
Therefore, the equation to find X (hypotenuse) will be:
\[
X^2 = 10^2 + 24^2
\]
\[
X^2 = 100 + 576
\]
\[
X^2 = 676
\]
4. **Solve for X**:
\[
X = \sqrt{676}
\]
\[
X = 26 \text{ units}
\]
Thus, the length of the hypotenuse (X) is 26 units.
### Summary
- Identified given side lengths of a right triangle (legs = 10 units and 24 units).
- Used the Pythagorean theorem to solve for the length of the hypotenuse.
- Found the hypotenuse length to be 26 units.
This problem illustrates the use of the Pythagorean theorem in solving for an unknown side of a right triangle, a fundamental concept in geometry.
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