Find the length of segment AC?
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Understanding Right Triangles: Finding the Length of a Segment
**Problem:**
Find the length of segment AC.
**Diagram Description:**
The diagram presents a right triangle with the right angle at vertex A. The vertices are labeled as follows:
- A is at the right angle
- B is at the end of the horizontal leg
- C is at the end of the vertical leg perpendicular to AB
The lengths of the sides are given as:
- AB = 5 units
- BC (the hypotenuse) = 13 units
- AC (the vertical leg) is the unknown length we need to find.
**Problem Solution:**
We are given a right triangle with the lengths of one leg (AB) and the hypotenuse (BC). To find the length of the other leg (AC), we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Here, \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. Let's denote:
- \(a = AC\) (the length we want to find)
- \(b = AB = 5\)
- \(c = BC = 13\)
Substituting the known values into the Pythagorean theorem, we get:
\[ AC^2 + 5^2 = 13^2 \]
\[ AC^2 + 25 = 169 \]
Subtract 25 from both sides:
\[ AC^2 = 144 \]
Taking the square root of both sides:
\[ AC = 12 \]
Therefore, the length of segment AC is 12 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa89ab128-ccb7-4301-9706-c15333e37929%2F99e4568d-a060-47c0-b3ba-102c5012d3d4%2F12cfolg.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Right Triangles: Finding the Length of a Segment
**Problem:**
Find the length of segment AC.
**Diagram Description:**
The diagram presents a right triangle with the right angle at vertex A. The vertices are labeled as follows:
- A is at the right angle
- B is at the end of the horizontal leg
- C is at the end of the vertical leg perpendicular to AB
The lengths of the sides are given as:
- AB = 5 units
- BC (the hypotenuse) = 13 units
- AC (the vertical leg) is the unknown length we need to find.
**Problem Solution:**
We are given a right triangle with the lengths of one leg (AB) and the hypotenuse (BC). To find the length of the other leg (AC), we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Here, \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. Let's denote:
- \(a = AC\) (the length we want to find)
- \(b = AB = 5\)
- \(c = BC = 13\)
Substituting the known values into the Pythagorean theorem, we get:
\[ AC^2 + 5^2 = 13^2 \]
\[ AC^2 + 25 = 169 \]
Subtract 25 from both sides:
\[ AC^2 = 144 \]
Taking the square root of both sides:
\[ AC = 12 \]
Therefore, the length of segment AC is 12 units.
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