Math / Geometry

Question4. Find the measure of $\angle B$ :-

Studdy Solution

STEP 1

1. The triangle is a right triangle, and $\angle C$ is the right angle.
2. The side lengths provided satisfy the Pythagorean theorem.
3. We need to find the measure of $\angle B$ .

STEP 2

1. Verify the triangle using the Pythagorean theorem.
2. Use trigonometric ratios to find $\angle B$ .

STEP 3

Verify the triangle using the Pythagorean theorem. For a right triangle, the sum of the squares of the two shorter sides should equal the square of the hypotenuse. Check:
$BC^2 + AC^2 = AB^2$ $5^2 + 12^2 = 13^2$ $25 + 144 = 169$ $169 = 169$
Since the equation holds true, the triangle is a valid right triangle.

STEP 4

Use trigonometric ratios to find $\angle B$ . Since we have the opposite side $BC = 5$ and the adjacent side $AC = 12$ , we can use the tangent function:
$\tan(\angle B) = \frac{\text{opposite}}{\text{adjacent}}$ $\tan(\angle B) = \frac{BC}{AC}$ $\tan(\angle B) = \frac{5}{12}$

STEP 5

To find $\angle B$ , take the arctangent (inverse tangent) of $\frac{5}{12}$ :
$\angle B = \tan^{-1}\left(\frac{5}{12}\right)$
Using a calculator, we find:
$\angle B \approx 22.62^\circ$
The measure of $\angle B$ is approximately:
$\boxed{22.62^\circ}$

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