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 35° 35° 55° 55° 70° 70° 90°
Answers
GPT-4o mini
In a circle, when a triangle is inscribed such that one side of the triangle is the diameter of the circle, the angle opposite that side (inscribed angle) is a right angle (90°). This is known as Thales' theorem.
In this case, since AB is the diameter of the circle and C is the vertex opposite the diameter, angle C is 90°.
To find angle B, we can use the triangle sum theorem, which states that the sum of the angles in a triangle is 180°. Therefore:
\[
\text{Angle A} + \text{Angle B} + \text{Angle C} = 180°
\]
Given that angle C = 90°, we can write:
\[
\text{Angle A} + \text{Angle B} + 90° = 180°
\]
This simplifies to:
\[
\text{Angle A} + \text{Angle B} = 90°
\]
Now, since we have the lengths of the sides \(AB\) (diameter, 14 cm) and \(AC\) (8 cm), we can find angle A using the sine rule or trigonometry.
1. We can find side \(BC\) using the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2 \implies 14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2 \implies BC^2 = 132 \implies BC \approx 11.48 \text{ cm}
\]
2. Now, we can find angle B using the cosine rule or trigonometric ratios. Since \(AB\) is the hypotenuse for triangle \(ABC\):
\[
\cos(A) = \frac{AC}{AB} = \frac{8}{14} \approx 0.5714
\]
Thus, angle A can be determined as follows:
\[
A \approx \cos^{-1}(0.5714) \approx 55°
\]
3. Now use the triangle sum theorem to find angle B:
\[
A + B + C = 180° \implies 55° + B + 90° = 180° \implies B = 180° - 145° = 35°
\]
Therefore, the approximate measure of angle B is **35°**.
In this case, since AB is the diameter of the circle and C is the vertex opposite the diameter, angle C is 90°.
To find angle B, we can use the triangle sum theorem, which states that the sum of the angles in a triangle is 180°. Therefore:
\[
\text{Angle A} + \text{Angle B} + \text{Angle C} = 180°
\]
Given that angle C = 90°, we can write:
\[
\text{Angle A} + \text{Angle B} + 90° = 180°
\]
This simplifies to:
\[
\text{Angle A} + \text{Angle B} = 90°
\]
Now, since we have the lengths of the sides \(AB\) (diameter, 14 cm) and \(AC\) (8 cm), we can find angle A using the sine rule or trigonometry.
1. We can find side \(BC\) using the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2 \implies 14^2 = 8^2 + BC^2
\]
\[
196 = 64 + BC^2 \implies BC^2 = 132 \implies BC \approx 11.48 \text{ cm}
\]
2. Now, we can find angle B using the cosine rule or trigonometric ratios. Since \(AB\) is the hypotenuse for triangle \(ABC\):
\[
\cos(A) = \frac{AC}{AB} = \frac{8}{14} \approx 0.5714
\]
Thus, angle A can be determined as follows:
\[
A \approx \cos^{-1}(0.5714) \approx 55°
\]
3. Now use the triangle sum theorem to find angle B:
\[
A + B + C = 180° \implies 55° + B + 90° = 180° \implies B = 180° - 145° = 35°
\]
Therefore, the approximate measure of angle B is **35°**.