Final answer:
The ratio of the area of the smaller square to the area of the circle is 1:π, given a circle radius of 10 cm and the smaller square vertices being the midpoints of the larger square.
Step-by-step explanation:
The question asks us to find the ratio of the area of a smaller square to the area of a circle, with the circle having a radius of 10 centimeters.
The vertices of the smaller square are the midpoints of the larger square's sides. To find the area of the circle, we use the formula A = πr². For a radius of 10 cm, the area of the circle is π × 10² = 100π cm².
Understanding that the larger square has a side equal to the diameter of the circle (20 cm), the smaller square will have sides half the length of the larger square, which means each side of the smaller square is 10 cm. Therefore, the area of the smaller square is 10² = 100 cm².
The ratio of the smaller square's area to the circle's area is 100 cm² (smaller square) to 100π cm² (circle), simplifying to 1:π. This represents the ratio of the area of the smaller square to the area of the circle, expressed as a common fraction in terms of π.