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° 90° 90° 70°

Answer 1

In a circle, when a triangle is inscribed such that one side of the triangle is the diameter, the angle opposite that diameter (in this case angle C) is a right angle (90°).

Using the triangle ABC, where AB is the diameter (14 cm) and AC is one of the sides (8 cm), we can find the length of BC using the Pythagorean theorem.

Given:
- AB (the diameter) = 14 cm
- AC = 8 cm
- Angle C = 90°

Let’s denote:
- BC = b
- AC = 8 cm
- AB = 14 cm (diameter)

The Pythagorean theorem states:
\[ AB^2 = AC^2 + BC^2 \]
\[ 14^2 = 8^2 + b^2 \]
\[ 196 = 64 + b^2 \]
\[ b^2 = 196 - 64 \]
\[ b^2 = 132 \]
\[ b = \sqrt{132} \approx 11.49 \text{ cm} \]

Now we can use the sine or cosine rule to find angle B. We can calculate using the sine function:
\[ \sin(B) = \frac{AC}{AB}, \text{ or } \sin(B) = \frac{8}{14} = \frac{4}{7}. \]

Now we can find angle B using the inverse sine function:
\[ B \approx \arcsin\left(\frac{4}{7}\right). \]

Evaluating this gives about:
\[ B \approx 35° \]

So, the approximate angle measure of B is **35°**.