In a triangle inscribed in a circle with one side as the diameter (AB), the angle opposite that side (angle C) is a right angle (90°) according to the inscribed angle theorem.
Given that AB is the diameter of the circle and measures 14 cm, and AC measures 8 cm, we need to determine the measure of angle B.
We can use the right triangle relationships, specifically the sine function:
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In triangle ABC, since angle C is 90°, we can label AB as the hypotenuse.
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We can use the definition of the sine of an angle:
\[ \sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7} \]
- Now we can use the arcsin function to find angle B:
\[ B \approx \arcsin\left(\frac{4}{7}\right) \approx 0.585 \text{ radians} \approx 33.5^\circ \]
However, we can also use the cosine function to find angle B:
\[ \cos(B) = \frac{BC}{AB} \]
To find BC, we can use the Pythagorean theorem since \( AB^2 = AC^2 + BC^2 \):
\[ 14^2 = 8^2 + BC^2 \Rightarrow 196 = 64 + BC^2 \Rightarrow BC^2 = 132 \Rightarrow BC \approx 11.49 \]
Now substituting in:
\[ \cos(B) \approx \frac{BC}{AB} = \frac{11.49}{14} \approx 0.820 \]
Now let’s find \( B \):
\[ B \approx \arccos\left(0.820\right) \approx 35^\circ \]
Thus, the approximate angle measure of B is:
35°.