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
70°
70°
90°
90°
35°
35°
55°
55°
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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
70°
70°
90°
90°
35°
35°
55°
55°
Skip to navigation
Answers
GPT-4o mini
To find the angle measure of angle \( B \) in triangle \( ABC \) where \( AB \) is the diameter of the circle, we can use the properties of inscribed angles.
According to the inscribed angle theorem, an angle inscribed in a semicircle (where the endpoints of the angle are on the circle and the vertex of the angle is on the circle directly above the diameter) is a right angle. Thus, since \( AB \) is the diameter, angle \( C \) would be \( 90^\circ \).
Next, since we have triangle \( ABC \), we can use the triangle angle sum property which states that the sum of angles in a triangle is \( 180^\circ \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
Given that \( \angle C = 90^\circ \), we have:
\[
\angle A + \angle B + 90^\circ = 180^\circ
\]
This simplifies to:
\[
\angle A + \angle B = 90^\circ
\]
Since \( AC \) is 8 cm and \( AB \) is 14 cm, we can use the sine rule or trigonometric relationships, but first we explore a reasonable estimation approach based on the triangle's proportions.
Using the known lengths:
- \( AB = 14 \) cm (hypotenuse, being the diameter)
- \( AC = 8 \) cm
Using the Pythagorean theorem to estimate \( BC \):
\[
BC = \sqrt{AB^2 - AC^2} = \sqrt{14^2 - 8^2} = \sqrt{196 - 64} = \sqrt{132} \approx 11.49 \text{ cm}
\]
Next, we can find \( \angle B \) using the sine function:
\[
\sin(\angle A) = \frac{AC}{AB} = \frac{8}{14} \approx 0.571
\]
Thus:
\[
\angle A \approx \arcsin(0.571) \approx 35^\circ
\]
So:
\[
\angle B \approx 90^\circ - 35^\circ = 55^\circ
\]
Therefore, the approximate measure of angle \( B \) is:
**55°**
According to the inscribed angle theorem, an angle inscribed in a semicircle (where the endpoints of the angle are on the circle and the vertex of the angle is on the circle directly above the diameter) is a right angle. Thus, since \( AB \) is the diameter, angle \( C \) would be \( 90^\circ \).
Next, since we have triangle \( ABC \), we can use the triangle angle sum property which states that the sum of angles in a triangle is \( 180^\circ \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
Given that \( \angle C = 90^\circ \), we have:
\[
\angle A + \angle B + 90^\circ = 180^\circ
\]
This simplifies to:
\[
\angle A + \angle B = 90^\circ
\]
Since \( AC \) is 8 cm and \( AB \) is 14 cm, we can use the sine rule or trigonometric relationships, but first we explore a reasonable estimation approach based on the triangle's proportions.
Using the known lengths:
- \( AB = 14 \) cm (hypotenuse, being the diameter)
- \( AC = 8 \) cm
Using the Pythagorean theorem to estimate \( BC \):
\[
BC = \sqrt{AB^2 - AC^2} = \sqrt{14^2 - 8^2} = \sqrt{196 - 64} = \sqrt{132} \approx 11.49 \text{ cm}
\]
Next, we can find \( \angle B \) using the sine function:
\[
\sin(\angle A) = \frac{AC}{AB} = \frac{8}{14} \approx 0.571
\]
Thus:
\[
\angle A \approx \arcsin(0.571) \approx 35^\circ
\]
So:
\[
\angle B \approx 90^\circ - 35^\circ = 55^\circ
\]
Therefore, the approximate measure of angle \( B \) is:
**55°**