Question

Both circle Q and circle R have a central angle measuring 75°. The ratio of circle Q's radius to circle R's radius is 2:5. Which ratio represents the area of the sector for circle R to the area of the sector for circle Q?

Answer 1

The ratio of similar areas is the square of the ratio of the scale factor.

Circle R's sector is (5/2)² = 25/4 the area of Circle Q's sector.

Answer 2

Answer:

The ratio of the area of the sector for circle R to the area of the sector for circle Q is 25:4.

Step-by-step explanation:

Given information: Q are R different circles. The ratio of circle Q's radius to circle R's radius is 2:5.

Let the radius of Q and R are 2x and 5x respectively.

The central angle of each circle is 75°.

The area of a sector is

$A=\pi r^2(\frac{\theta}{360^{\circ}})$

where, r is the radius of circle and θ is central angle of sector.

Area of sector of circle Q.

$A_Q=\pi (2x)^2(\frac{75^{\circ}}{360^{\circ}})$

Area of sector of circle R.

$A_R=\pi (5x)^2(\frac{75^{\circ}}{360^{\circ}})$

The ratio of area of sector for circle R to the area of the sector for circle Q is

$\frac{A_R}{A_Q}=\frac{\pi (5x)^2(\frac{75^{\circ}}{360^{\circ}})}{\pi (2x)^2(\frac{75^{\circ}}{360^{\circ}})}$

$\frac{A_R}{A_Q}=\frac{25x^2}{4x^2}$

$\frac{A_R}{A_Q}=\frac{25}{4}$

Therefore the ratio of the area of the sector for circle R to the area of the sector for circle Q is 25:4.