Question

Regina draws a triangle inside a circle; she labels the vertices ABC, with the diameter as AB and the inscribed angle as C. She draws the length of AB 14 cm long, and AC 8 cm long. Using what you know about inscribed angles, what will be the approximate angle measure of B?

Answer 1

To find the angle B in triangle ABC inscribed in a circle, where AB is the diameter, we can use the fact that angle C is an inscribed angle that intercepts the diameter (AB). According to the inscribed angle theorem, any angle inscribed in a semicircle is a right angle (90 degrees). Hence, angle C is 90 degrees.

Next, we can use the lengths of sides AC (8 cm) and AB (14 cm) to find the length of side BC using the Pythagorean theorem. Let us denote:
- AB = c = 14 cm (the diameter)
- AC = b = 8 cm
- BC = a (the length we need to find)

In a right triangle:
\[
a^2 + b^2 = c^2
\]

Substituting the known lengths:
\[
a^2 + 8^2 = 14^2
\]
\[
a^2 + 64 = 196
\]
\[
a^2 = 196 - 64
\]
\[
a^2 = 132
\]
\[
a = \sqrt{132} \approx 11.49 \text{ cm}
\]

Now we can use the sine rule or trigonometric ratios to find angle B. In triangle ABC, we know:
\[
\sin C = \frac{b}{c} = \frac{AC}{AB} = \frac{8}{14}
\]

Now that we have angle C as 90 degrees, we can find angle B using the cosine rule:
\[
\cos B = \frac{AC}{AB} = \frac{8}{14}
\]

To find angle B, we need to calculate the angle whose cosine is \(\frac{8}{14}\):
\[
B \approx \cos^{-1}\left(\frac{8}{14}\right) \approx \cos^{-1}(0.5714)
\]

Using a calculator,
\[
B \approx 55.0 \text{ degrees}
\]

Thus, the approximate measure of angle B is:
\[
\boxed{55} \text{ degrees}
\]