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 ?(1 point)
Responses

35°
35°

55°
55°

70°
70°

90°

Answer 1

In a circle, if a triangle is inscribed such that one of its sides is the diameter of the circle, then the angle opposite to that side is a right angle (90°). This is known as the Inscribed Angle Theorem.

In Regina's triangle ABC, since AB is the diameter of the circle, angle C (the inscribed angle) must be 90°.

To find angle B, we can use the relationship between the sides and angles in triangle ABC. We know that:

- The length of AB (the hypotenuse) is 14 cm.
- The length of AC is 8 cm.

Using the Pythagorean theorem in triangle ABC, we can find BC:
\[
AB^2 = AC^2 + BC^2
\]
\[
14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2
\]
\[
BC^2 = 196 - 64 = 132
\]
\[
BC = \sqrt{132} \approx 11.49 \text{ cm}
\]

Now we can find angle B using the sine function:
\[
\sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7} \approx 0.571
\]

Now, find angle B:
\[
B \approx \sin^{-1}(0.571) \approx 35°
\]

Thus, the approximate angle measure of B is 35°.

The correct response is:
**35°**