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°

35°
35°

90°
90°

70°

Answer 1

Since triangle ABC is inscribed in a circle with AB as the diameter, angle C is a right angle according to the inscribed angle theorem (which states that an angle inscribed in a semicircle is a right angle).

Given that AB (the diameter) is 14 cm and AC is 8 cm, we can apply the Pythagorean theorem to find the length of BC and then use that to find the angle at B.

1. **Find length of BC using the Pythagorean theorem:**
\[
AB^2 = AC^2 + BC^2
\]
where \( AB = 14 \) cm and \( AC = 8 \) cm.
\[
14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2
\]
\[
BC^2 = 196 - 64 = 132
\]
\[
BC = \sqrt{132} \approx 11.49 \text{ cm}
\]

2. **Now, finding angle B:**
We can use trigonometric ratios, specifically the sine function, since we have the lengths of the opposite side (AC) and the hypotenuse (AB).
\[
\sin B = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7}
\]

3. **Calculating angle B:**
To find angle B, we take the inverse sine:
\[
B \approx \sin^{-1} \left(\frac{4}{7}\right)
\]

Calculating this gives approximately:
\[
B \approx 36.87^\circ
\]

Thus, the closest answer from the options you provided is **35°**.