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°
70°
Responses
90°
90°
55°
55°
35°
35°
70°
70°
Answers
GPT-4o mini
To find the angle \( B \) in triangle \( ABC \), we note that since \( AB \) is the diameter of the circle, angle \( C \) is a right angle (90°), according to the inscribed angle theorem.
Given:
- \( AB = 14 \, \text{cm} \)
- \( AC = 8 \, \text{cm} \)
Using the Pythagorean theorem for right triangle \( ABC \):
\[
AB^2 = AC^2 + BC^2
\]
Substituting the known values:
\[
14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2
\]
\[
BC^2 = 196 - 64
\]
\[
BC^2 = 132
\]
\[
BC = \sqrt{132} \approx 11.49 \, \text{cm}
\]
Now, we can use the definition of the sine function to find angle \( B \):
\[
\sin B = \frac{AC}{AB}
\]
Substituting the known lengths:
\[
\sin B = \frac{8}{14} = \frac{4}{7}
\]
Now we calculate \( B \):
\[
B \approx \arcsin\left(\frac{4}{7}\right)
\]
Using a calculator, we find:
\[
B \approx 46.57^\circ
\]
However, we need to find angle \( A \) using the angle sum property of triangles:
\[
A + B + C = 180^\circ
\]
Substituting \( C = 90^\circ \):
\[
A + B + 90^\circ = 180^\circ
\]
\[
A + B = 90^\circ
\]
Thus,
\[
A = 90^\circ - B \approx 90^\circ - 46.57^\circ \approx 43.43^\circ
\]
Finally, we can find angle \( B \) still:
Knowing \( A + B = 90^\circ \), thus angle \( B \) remains around 46.57°, close to selected choices provided.
The response closest and most reasonable choice for an approximate angle measure for angle \( B \) could then be considered to align with what angles are available. However, since this approach aligns with approximate number rounding:
The options presented in the problem do not contain 46.57° but among the options arranged, response \( 55° \) seems to be the most plausible leaning direction when rounding, as approximate can skew higher.
Thus,
**Response: 55°**
Given:
- \( AB = 14 \, \text{cm} \)
- \( AC = 8 \, \text{cm} \)
Using the Pythagorean theorem for right triangle \( ABC \):
\[
AB^2 = AC^2 + BC^2
\]
Substituting the known values:
\[
14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2
\]
\[
BC^2 = 196 - 64
\]
\[
BC^2 = 132
\]
\[
BC = \sqrt{132} \approx 11.49 \, \text{cm}
\]
Now, we can use the definition of the sine function to find angle \( B \):
\[
\sin B = \frac{AC}{AB}
\]
Substituting the known lengths:
\[
\sin B = \frac{8}{14} = \frac{4}{7}
\]
Now we calculate \( B \):
\[
B \approx \arcsin\left(\frac{4}{7}\right)
\]
Using a calculator, we find:
\[
B \approx 46.57^\circ
\]
However, we need to find angle \( A \) using the angle sum property of triangles:
\[
A + B + C = 180^\circ
\]
Substituting \( C = 90^\circ \):
\[
A + B + 90^\circ = 180^\circ
\]
\[
A + B = 90^\circ
\]
Thus,
\[
A = 90^\circ - B \approx 90^\circ - 46.57^\circ \approx 43.43^\circ
\]
Finally, we can find angle \( B \) still:
Knowing \( A + B = 90^\circ \), thus angle \( B \) remains around 46.57°, close to selected choices provided.
The response closest and most reasonable choice for an approximate angle measure for angle \( B \) could then be considered to align with what angles are available. However, since this approach aligns with approximate number rounding:
The options presented in the problem do not contain 46.57° but among the options arranged, response \( 55° \) seems to be the most plausible leaning direction when rounding, as approximate can skew higher.
Thus,
**Response: 55°**