Question
In \( \triangle \mathrm{EFG} \), the measure of \( \angle \mathrm{G}=90^{\circ}, \mathrm{EG}=11, \mathrm{GF}=60 \), and \( \mathrm{FE}=61 \). What ratio represents the tangent of \( \angle \mathrm{E} \) ?
Ask by Carter Barnett. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The tangent of angle E is 60 to 11, or tan E = 60 : 11
Solution
In \( \triangle \mathrm{EFG} \), angle \( \mathrm{G} \) is the right angle (\( 90^\circ \)). The sides are given as:
- \( \mathrm{EG} = 11 \)
- \( \mathrm{GF} = 60 \)
- \( \mathrm{FE} = 61 \)
Since \( \angle \mathrm{G} \) is the right angle, the sides \( \mathrm{EG} \) and \( \mathrm{GF} \) are the legs of the triangle, and \( \mathrm{FE} \) is the hypotenuse.
To find \( \tan(\angle \mathrm{E}) \), we use the definition of tangent in a right triangle:
\[
\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}
\]
Here, for \( \angle \mathrm{E} \):
- The opposite side is \( \mathrm{GF} = 60 \)
- The adjacent side is \( \mathrm{EG} = 11 \)
Therefore:
\[
\tan(\angle \mathrm{E}) = \frac{60}{11} \quad \text{or} \quad 60:11
\]
**Answer:** The tangent of angle E is 60 to 11, that is, tan E = 60 : 11
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Simplify this solution Beyond the Answer
To find the tangent of \( \angle E \) in triangle \( \triangle EFG \), we can use the definition of tangent in a right triangle, which is the ratio of the length of the opposite side to the length of the adjacent side. In \( \triangle EFG \): - \( \angle G = 90^\circ \) - The length of side \( EF \) is opposite \( \angle E \). - The length of side \( EG \) is adjacent to \( \angle E \). Given: - \( EG = 11 \) (adjacent side) - \( EF = 61 \) (opposite side) So, the tangent of \( \angle E \) is given by: \[ \tan E = \frac{\text{opposite}}{\text{adjacent}} = \frac{EF}{EG} = \frac{61}{11}. \] Thus, the ratio that represents the tangent of \( \angle E \) is \( \frac{61}{11} \).
