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)

Answer 1

In a circle, an inscribed angle is half of the measure of the central angle that subtends the same arc. Since \(AB\) is the diameter of the circle, the angle \(C\) at vertex \(C\) is a right angle (90 degrees) because of the Thales' theorem.

To find the angle \(B\), we can use the properties of triangles and the relationship between the sides and angles. Given that we have a right triangle \(ABC\) with \(C\) being the right angle:

1. We know \(AB = 14\) cm (the hypotenuse),
2. \(AC = 8\) cm (one of the legs).

We can find side \(BC\) using the Pythagorean theorem:

\[
AB^2 = AC^2 + BC^2
\]

Substituting the values:

\[
14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2
\]
\[
BC^2 = 196 - 64
\]
\[
BC^2 = 132
\]
\[
BC \approx \sqrt{132} \approx 11.49 \text{ cm}
\]

Now, we want to find angle \(B\). We can use the sine function:

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

To find angle \(B\):

\[
B \approx \sin^{-1}\left(\frac{4}{7}\right)
\]

Calculating this gives:

\[
B \approx 0.5932 \text{ radians} \approx 34.03 \text{ degrees}
\]

Therefore, the approximate measure of angle \(B\) is **34 degrees**.