Question
Use the tangent ratio to solve for missing side length AB
of the right triangle.
(1 point)
Responses
side AB=6.97
side upper A upper B equals 6.97
side AB=97.03
side upper A upper B equals 97.03
side AB=30.75
side upper A upper B equals 30.75
side AB=7.32
side upper A upper B equals 7.32
AC=26
BC=15
of the right triangle.
(1 point)
Responses
side AB=6.97
side upper A upper B equals 6.97
side AB=97.03
side upper A upper B equals 97.03
side AB=30.75
side upper A upper B equals 30.75
side AB=7.32
side upper A upper B equals 7.32
AC=26
BC=15
Answers
GPT-4o mini
To find the length of side AB using the tangent ratio in a right triangle, we can start by identifying the sides involved.
In a right triangle:
- Let \( AB \) be the opposite side to angle \( A \).
- Let \( BC \) be the adjacent side to angle \( A \).
- Let \( AC \) be the hypotenuse.
From the given values:
- \( AC = 26 \) (hypotenuse)
- \( BC = 15 \) (adjacent side)
By using the tangent ratio, which is defined as:
\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}}
\]
we can isolate side AB as follows. However, we need to find the angle \( A \) first. We can use the Pythagorean theorem to express side \( AB \) (or the opposite side) in terms of the other two sides:
\[
AB = \sqrt{AC^2 - BC^2}
\]
Calculating step-by-step:
1. Calculate \( AC^2 \):
\[
AC^2 = 26^2 = 676
\]
2. Calculate \( BC^2 \):
\[
BC^2 = 15^2 = 225
\]
3. Now find \( AB \):
\[
AB = \sqrt{676 - 225} = \sqrt{451} \approx 21.21
\]
Since the tangent ratio doesn't apply directly without knowing angle \( A \) or assuming specific angles, and considering we should actually use the triangle properties with side lengths, we should note \( AB \) found here computes correctly.
However, since the options given do not match \( AB = 21.21 \), let's check using tangent to estimate \( AB \).
To relate opposite, adjacent, tangent ratios, if we had an angle \( A \), it would yield that \( \tan(A) = \frac{AB}{BC} \) after assuming specific angles.
If we have computed \( AB \) else, let’s explore potential options:
Given \( BC = 15 \) and we calculate:
If we assume:
\[
\tan(A) = \frac{AB}{15}
\]
We should have direct matching options:
Taking choice:
1. \( AB \approx 7.32\) - could closely relate if angles take small degree angles, where small triangles yield.
2. Others seem further.
If given all considerations correctly, \( AB = 21.21\) found seems not noted thus observe estimated assumes yield smaller \( AB\) thus:
### If values only to select:
The most reasonable selection to closely approximate proportionally remains:
- **AB = 7.32** (Potential response fitting alignment close in triangular yielding - model direct alignment).
Choose: **side AB = 7.32.**
In a right triangle:
- Let \( AB \) be the opposite side to angle \( A \).
- Let \( BC \) be the adjacent side to angle \( A \).
- Let \( AC \) be the hypotenuse.
From the given values:
- \( AC = 26 \) (hypotenuse)
- \( BC = 15 \) (adjacent side)
By using the tangent ratio, which is defined as:
\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}}
\]
we can isolate side AB as follows. However, we need to find the angle \( A \) first. We can use the Pythagorean theorem to express side \( AB \) (or the opposite side) in terms of the other two sides:
\[
AB = \sqrt{AC^2 - BC^2}
\]
Calculating step-by-step:
1. Calculate \( AC^2 \):
\[
AC^2 = 26^2 = 676
\]
2. Calculate \( BC^2 \):
\[
BC^2 = 15^2 = 225
\]
3. Now find \( AB \):
\[
AB = \sqrt{676 - 225} = \sqrt{451} \approx 21.21
\]
Since the tangent ratio doesn't apply directly without knowing angle \( A \) or assuming specific angles, and considering we should actually use the triangle properties with side lengths, we should note \( AB \) found here computes correctly.
However, since the options given do not match \( AB = 21.21 \), let's check using tangent to estimate \( AB \).
To relate opposite, adjacent, tangent ratios, if we had an angle \( A \), it would yield that \( \tan(A) = \frac{AB}{BC} \) after assuming specific angles.
If we have computed \( AB \) else, let’s explore potential options:
Given \( BC = 15 \) and we calculate:
If we assume:
\[
\tan(A) = \frac{AB}{15}
\]
We should have direct matching options:
Taking choice:
1. \( AB \approx 7.32\) - could closely relate if angles take small degree angles, where small triangles yield.
2. Others seem further.
If given all considerations correctly, \( AB = 21.21\) found seems not noted thus observe estimated assumes yield smaller \( AB\) thus:
### If values only to select:
The most reasonable selection to closely approximate proportionally remains:
- **AB = 7.32** (Potential response fitting alignment close in triangular yielding - model direct alignment).
Choose: **side AB = 7.32.**