Question

Use the inverse of the tangent ratio to find the approximate angle measure at vertex B . Round your answer to the nearest whole degree. ( It's an ABC triangle. It does have a right angle. We don't know what AB equals. BC equals 20, CA equals 25)

Answer 1

To find the angle measure at vertex B in triangle ABC, we can use the inverse tangent (arctan) function. Since triangle ABC has a right angle, we can identify which angles (A, B, C) correspond to which sides.

In this triangle:
- We know that side BC is opposite angle A, which measures 20.
- Side CA is hypotenuse, which measures 25.
- Side AB is adjacent to angle B, which we'll denote as \( x \).

To find angle B, we will use the relationship given by the tangent function:

\[
\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{AB}
\]

However, we don’t have AB yet. But we can use the Pythagorean theorem to find AB:

\[
AC^2 = AB^2 + BC^2
\]

Substituting the lengths we know:

\[
25^2 = AB^2 + 20^2
\]

Calculating the squares:

\[
625 = AB^2 + 400
\]

Now, solving for \( AB^2 \):

\[
AB^2 = 625 - 400 = 225
\]

Taking the square root gives:

\[
AB = \sqrt{225} = 15
\]

Now that we have the lengths of all sides, we can find angle B using the tangent ratio:

\[
\tan(B) = \frac{BC}{AB} = \frac{20}{15}
\]

Calculating \( \tan(B) \):

\[
\tan(B) = \frac{20}{15} = \frac{4}{3}
\]

Now we find angle B using the inverse tangent:

\[
B = \tan^{-1}\left(\frac{4}{3}\right)
\]

Using a calculator to find the angle:

\[
B \approx 53.13 \text{ degrees}
\]

Rounding to the nearest whole degree, the angle measure at vertex B is:

\[
\boxed{53} \text{ degrees}
\]