Question

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

90°
90°

55°
55°

35°
35°

70°

Answer 1

In this scenario, since triangle ABC is inscribed in a circle with AB as the diameter, angle C is a right angle according to Thales' theorem. Thus, angle C measures 90°.

To find the angle measure of angle B, we can use the properties of triangles. We know the following:

1. The sum of the angles in a triangle is 180°.
2. We have angle C = 90° (right angle).
3. The angle measures for angles A and B can be calculated as follows:

\[
\text{Angle A} + \text{Angle B} + \text{Angle C} = 180°
\]

Substituting angle C:

\[
\text{Angle A} + \text{Angle B} + 90° = 180°
\]

This simplifies to:

\[
\text{Angle A} + \text{Angle B} = 90°
\]

Since we know that AC is 8 cm and AB (the diameter) is 14 cm, we can use the sine rule to find the angles, but we primarily need angle B.

Using the sine rule relates the sides opposite to the angles:

\[
\frac{AC}{\sin B} = \frac{AB}{\sin C}
\]

Since angle C is 90°, \(\sin C = 1\):

\[
\frac{8}{\sin B} = \frac{14}{1} \implies \sin B = \frac{8}{14} = \frac{4}{7}
\]

Now, we calculate angle B:

Using that \(\sin B \approx 0.5714\).

To find angle B, we can use a calculator:

\[
B \approx \sin^{-1}(0.5714) \approx 35°
\]

Thus, the approximate measure of angle B is:

**35°**.