Duplicate Question
The question on this page has been marked as a duplicate question.
Original Question
Regina draws a triangle inside a circle; she labels the vertices ABC , with the diameter as AB and the inscribed angle as C . S...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°
55°
55°
35°
35°
90°
, 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°
55°
55°
35°
35°
90°
Answers
GPT-4o mini
In a circle, if a triangle is inscribed with one side as the diameter (in this case \( AB \)), then the angle opposite that side (the inscribed angle \( C \) in triangle \( ABC \)) is always a right angle. This is known from the inscribed angle theorem which states that an inscribed angle subtended by a diameter is 90 degrees.
Since \( C \) is the angle opposite the diameter \( AB \), we can conclude that:
\[
\angle C = 90^\circ
\]
Now, we know that the sum of angles in a triangle is \( 180^\circ \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
If \( \angle C = 90^\circ \), then:
\[
\angle A + \angle B = 180^\circ - 90^\circ = 90^\circ
\]
To estimate the approximate measure of \( \angle B \), we can use the lengths of the sides:
- \( AC = 8 \, \text{cm} \)
- \( AB = 14 \, \text{cm} \)
Assuming that \( AC \) and \( AB \) form a right triangle with \( C \) at the right angle, we can use the sine rule or cosine rule to find angles \( A \) and \( B \). However, since we are asked only for \( \angle B \) and not for the exact calculations involving \( \angle A \) and \( \angle B \) individually, we can respond based on the options given.
Given the information, we can infer that:
\[
\angle B + \angle A = 90^\circ
\]
Hence, \( \angle B \) must be less than \( 90^\circ \).
From the possible answers:
- 70°
- 55°
- 35°
- 90°
The two options 70° and 55° are the only reasonable angles for \( \angle B \) in a right triangle configuration (where \( 90^\circ \geq \angle B > 0^\circ \)).
Considering \( AC < AB \), angle \( A \) should be greater than angle \( B \) (since opposite the longer side is the larger angle).
Thus, we can make a reasonable estimation that angle \( B \) is approximately:
\[
\text{Angle } B \approx 70^\circ
\]
So, the best response from the options given is:
**70°**.
Since \( C \) is the angle opposite the diameter \( AB \), we can conclude that:
\[
\angle C = 90^\circ
\]
Now, we know that the sum of angles in a triangle is \( 180^\circ \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
If \( \angle C = 90^\circ \), then:
\[
\angle A + \angle B = 180^\circ - 90^\circ = 90^\circ
\]
To estimate the approximate measure of \( \angle B \), we can use the lengths of the sides:
- \( AC = 8 \, \text{cm} \)
- \( AB = 14 \, \text{cm} \)
Assuming that \( AC \) and \( AB \) form a right triangle with \( C \) at the right angle, we can use the sine rule or cosine rule to find angles \( A \) and \( B \). However, since we are asked only for \( \angle B \) and not for the exact calculations involving \( \angle A \) and \( \angle B \) individually, we can respond based on the options given.
Given the information, we can infer that:
\[
\angle B + \angle A = 90^\circ
\]
Hence, \( \angle B \) must be less than \( 90^\circ \).
From the possible answers:
- 70°
- 55°
- 35°
- 90°
The two options 70° and 55° are the only reasonable angles for \( \angle B \) in a right triangle configuration (where \( 90^\circ \geq \angle B > 0^\circ \)).
Considering \( AC < AB \), angle \( A \) should be greater than angle \( B \) (since opposite the longer side is the larger angle).
Thus, we can make a reasonable estimation that angle \( B \) is approximately:
\[
\text{Angle } B \approx 70^\circ
\]
So, the best response from the options given is:
**70°**.