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 35° 35° 90° 90° 70° 70° 55°

Answer 1

To find the angle B in triangle ABC inscribed in a circle with AB as the diameter, we can use the fact that an inscribed angle (angle C) that subtends a diameter (AB) is a right angle. Therefore, angle C is 90°.

Next, we can apply the Law of Cosines or consider the known lengths of sides AC and AB to approximate angle B. However, since we only need the approximate value, we can focus on the triangle’s properties related to the sides.

Given:

• AB = 14 cm (the diameter)
• AC = 8 cm

In triangle ABC, we know:

• Angle C = 90° (because it subtends the diameter)

To find the angles in a right triangle, we can use basic trigonometric ratios. For angle B, we can use the sine or cosine function:

• $B{C}^2=A{B}^2-A{C}^2={14}^2-8^2=196-64=132$

Therefore, the length of side BC can be calculated as:

• $BC=\sqrt{132}\approx 11.49$ cm.

Next, we can use the sine function to find angle B:

• $\sin (B)=\frac{AC}{AB}=\frac{8}{14}=\frac{4}{7}$

Now calculate $B$:

• $B\approx \arcsin \left(\frac{4}{7}\right)\approx 0.588$ rad or approximately $33.69°$.

Based on the given choices, the closest and most reasonable approximations we see are not exactly there, but since B is around 35°, that would be the best answer.

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