Question

texes mathematics 4-8 (115)

1. in the following diagram, the area of the circle is (25pi).

what is the length of (\overline{gf})?

a. (\overline{gf} = 3)
b. (\overline{gf} = 6)
c. (\overline{gf} = 10)
d. (\overline{gf} = 8)

Explanation:

⚡ Using what you learned: right triangle trigonometry (soh cah toa) · area of circles and sectors

Step 1: Find the radius and diameter of the circle

The area of the circle is given as \( 25\pi \).

$$ \text{Area} = \pi r^2 = 25\pi \implies r^2 = 25 \implies r = 5 $$

Since the radius \( r = 5 \), the diameter of the circle is:

$$ d = 2r = 10 $$

The inscribed angle \( \angle HGF \) is a right angle (\( 90^\circ \)). An inscribed angle of \( 90^\circ \) must subtend a diameter. Therefore, the hypotenuse \( HF \) is the diameter of the circle:

$$ HF = 10 $$

Step 2: Calculate the length of segment GF

In the right triangle \( \triangle HGF \), the angle at \( H \) is given as \( 53^\circ \) (as indicated by the handwritten notes on the diagram, or we can use the trigonometric ratio for the opposite side).

Using the sine ratio for \( \angle H \):

$$ \sin(H) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{GF}{HF} $$

$$ \sin(53^\circ) \approx 0.8 $$

$$ 0.8 = \frac{GF}{10} \implies GF = 10 \times 0.8 = 8 $$

Answer:

D. \( \overline{GF} = 8 \)