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°
Answers
GPT-4o mini
To find the measure of angle B in the triangle ABC inscribed in a circle where AB is the diameter, we can use the property of inscribed angles. According to the inscribed angle theorem, an angle inscribed in a semicircle is a right angle. Since AB is the diameter, angle C (which is opposite to the diameter) is 90°.
Now, we know the following:
- AB (the diameter) = 14 cm
- AC = 8 cm
Since triangle ABC is a right triangle at C, we can use the Pythagorean theorem to find the length of BC.
Let \( BC = x \), then according to the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2
\]
\[
14^2 = 8^2 + x^2
\]
\[
196 = 64 + x^2
\]
\[
x^2 = 196 - 64 = 132
\]
\[
x = \sqrt{132} \approx 11.49 \text{ cm}
\]
Now, to find angle B, we can use the sine, cosine, or tangent ratios. Using the sine ratio:
\[
\sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7}
\]
To find the angle B, we calculate:
\[
B \approx \sin^{-1}\left(\frac{4}{7}\right)
\]
Calculating this gives approximately:
\[
B \approx 45.57^\circ
\]
However, angle A can be calculated as follows:
\[
\sin(A) = \frac{BC}{AB} = \frac{\sqrt{132}}{14} \approx \frac{11.49}{14}
\]
Finding angle A should also give us a rough idea about angle B given that in a triangle, the sum of angles is 180°. Thus, we approximate angle A and subsequently find B as follows:
\[
B + A + C = 180^\circ
\]
Given that angle C is 90°, we know:
\[
A + B = 90^\circ
\]
Estimating angle A from working towards \( B \approx 45^\circ \):
Thus, the apparent measures given in the responses would indicate:
- To approximate B as around 55°, 70°, or similar could mean optioning between given numbers rounding off from calculated tuples.
From your choices, the closest answer would be **55°** or **70°**. However, recognizing right triangle constraints makes 90° impossible for B. Thus, ideally, it would fall between those two options.
Overall, rounding to evaluated options suggests **55°** would align closest based on known rotational angle measures from triangle function ratios or estimated referenced construction of concept diametre. Hence, I would select **55°** as the answer reflecting back through evaluated ratios collectively.
Now, we know the following:
- AB (the diameter) = 14 cm
- AC = 8 cm
Since triangle ABC is a right triangle at C, we can use the Pythagorean theorem to find the length of BC.
Let \( BC = x \), then according to the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2
\]
\[
14^2 = 8^2 + x^2
\]
\[
196 = 64 + x^2
\]
\[
x^2 = 196 - 64 = 132
\]
\[
x = \sqrt{132} \approx 11.49 \text{ cm}
\]
Now, to find angle B, we can use the sine, cosine, or tangent ratios. Using the sine ratio:
\[
\sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7}
\]
To find the angle B, we calculate:
\[
B \approx \sin^{-1}\left(\frac{4}{7}\right)
\]
Calculating this gives approximately:
\[
B \approx 45.57^\circ
\]
However, angle A can be calculated as follows:
\[
\sin(A) = \frac{BC}{AB} = \frac{\sqrt{132}}{14} \approx \frac{11.49}{14}
\]
Finding angle A should also give us a rough idea about angle B given that in a triangle, the sum of angles is 180°. Thus, we approximate angle A and subsequently find B as follows:
\[
B + A + C = 180^\circ
\]
Given that angle C is 90°, we know:
\[
A + B = 90^\circ
\]
Estimating angle A from working towards \( B \approx 45^\circ \):
Thus, the apparent measures given in the responses would indicate:
- To approximate B as around 55°, 70°, or similar could mean optioning between given numbers rounding off from calculated tuples.
From your choices, the closest answer would be **55°** or **70°**. However, recognizing right triangle constraints makes 90° impossible for B. Thus, ideally, it would fall between those two options.
Overall, rounding to evaluated options suggests **55°** would align closest based on known rotational angle measures from triangle function ratios or estimated referenced construction of concept diametre. Hence, I would select **55°** as the answer reflecting back through evaluated ratios collectively.