Question
A right triangle is shown. The diagonal line measures 20, the horizontal line measures 25. A measurement is not given for the slanted line. Use the inverse of the tangent ratio to find the approximate angle measure at vertex B . Round your answer to the nearest whole degree. (1 point) θ≈ ° Skip to navigation
Answers
Answered by
GPT-4o mini
To find the angle measure at vertex B in the right triangle, you can use the tangent ratio:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, we can identify the lengths of the sides. The diagonal line (which is the hypotenuse) measures 20, and the horizontal line (adjacent to angle B) measures 25. The side opposite angle B can be found using the Pythagorean theorem:
Let \( a \) be the length of the side opposite angle B.
\[
a^2 + 25^2 = 20^2
\]
Calculating \( 25^2 \) and \( 20^2 \):
\[
a^2 + 625 = 400
\]
Now, solve for \( a^2 \):
\[
a^2 = 400 - 625
\]
\[
a^2 = -225
\]
Since this results in a negative value, it indicates there might be confusion in identifying the given values in the triangle. For a right triangle, the hypotenuse must always be the longest side. Since 25 > 20, we should assume the horizontal line is actually in the wrong position regarding the triangle’s properties.
However, if we were to find the angle using the right approach and correctly assign the sides regarding the correct triangle setup, it might be suitable to use:
Assume we want to find the angle where:
\[
\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)
\]
Given the lengths, assuming side opposite angle B measures 20 and the side adjacent measures 25:
Calculate:
\[
\tan(\theta) = \frac{20}{25}
\]
This simplifies to:
\[
\tan(\theta) = 0.8
\]
Now use the inverse tangent function to find \( \theta \):
\[
\theta = \tan^{-1}(0.8)
\]
Using a calculator:
\[
\theta \approx 38.66 \text{ degrees}
\]
Rounding to the nearest whole degree gives:
\[
\theta \approx 39 \text{ degrees}
\]
Thus, the approximate angle measure at vertex B is \( \theta \approx 39^\circ \).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, we can identify the lengths of the sides. The diagonal line (which is the hypotenuse) measures 20, and the horizontal line (adjacent to angle B) measures 25. The side opposite angle B can be found using the Pythagorean theorem:
Let \( a \) be the length of the side opposite angle B.
\[
a^2 + 25^2 = 20^2
\]
Calculating \( 25^2 \) and \( 20^2 \):
\[
a^2 + 625 = 400
\]
Now, solve for \( a^2 \):
\[
a^2 = 400 - 625
\]
\[
a^2 = -225
\]
Since this results in a negative value, it indicates there might be confusion in identifying the given values in the triangle. For a right triangle, the hypotenuse must always be the longest side. Since 25 > 20, we should assume the horizontal line is actually in the wrong position regarding the triangle’s properties.
However, if we were to find the angle using the right approach and correctly assign the sides regarding the correct triangle setup, it might be suitable to use:
Assume we want to find the angle where:
\[
\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)
\]
Given the lengths, assuming side opposite angle B measures 20 and the side adjacent measures 25:
Calculate:
\[
\tan(\theta) = \frac{20}{25}
\]
This simplifies to:
\[
\tan(\theta) = 0.8
\]
Now use the inverse tangent function to find \( \theta \):
\[
\theta = \tan^{-1}(0.8)
\]
Using a calculator:
\[
\theta \approx 38.66 \text{ degrees}
\]
Rounding to the nearest whole degree gives:
\[
\theta \approx 39 \text{ degrees}
\]
Thus, the approximate angle measure at vertex B is \( \theta \approx 39^\circ \).
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