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

55°
55°

70°
70°

35°
35°

90°

Answer 1

In a triangle inscribed in a circle with one side as the diameter (AB), the angle opposite that side (angle C) is a right angle (90°) according to the inscribed angle theorem.

Given that AB is the diameter of the circle and measures 14 cm, and AC measures 8 cm, we need to determine the measure of angle B.

We can use the right triangle relationships, specifically the sine function:

1. In triangle ABC, since angle C is 90°, we can label AB as the hypotenuse.

2. We can use the definition of the sine of an angle:

\[
\sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7}
\]

3. Now we can use the arcsin function to find angle B:

\[
B \approx \arcsin\left(\frac{4}{7}\right) \approx 0.585 \text{ radians} \approx 33.5^\circ
\]

However, we can also use the cosine function to find angle B:

\[
\cos(B) = \frac{BC}{AB}
\]

To find BC, we can use the Pythagorean theorem since \( AB^2 = AC^2 + BC^2 \):

\[
14^2 = 8^2 + BC^2 \Rightarrow 196 = 64 + BC^2 \Rightarrow BC^2 = 132 \Rightarrow BC \approx 11.49
\]

Now substituting in:

\[
\cos(B) \approx \frac{BC}{AB} = \frac{11.49}{14} \approx 0.820
\]

Now let’s find \( B \):

\[
B \approx \arccos\left(0.820\right) \approx 35^\circ
\]

Thus, the approximate angle measure of B is:

**35°**.