Question

PQRS is a square. G is the midpoint of PQ and H is the midpoint of QR. Triangle GHS is shaded. G Q H R not to scale S P. Find the ratio shaded area: area of the square in its simplest form. Show all your working.

A) 1:2
B) 1:3
C) 1:4
D) 1:5

Answer 1

The shaded area of triangle GHS is one-fourth of the area of square PQRS, resulting in a ratio of 1:4 for the shaded area to the square's area.

Step-by-step explanation:

To find the ratio of the shaded area to the area of the square, we first need to calculate the area of both the square and the shaded triangle GHS. We assume the square's side length is a; hence, the area of the square (PQRS) is a².

Since G is the midpoint of PQ and H is the midpoint of QR, the triangle base GH is equal to half the square's side (a/2) and the height HS is equal to the full side of the square (a). Using the formula for the area of a triangle, 1/2 × base × height, we find the area of triangle GHS to be:

Area of GHS = 1/2 × (a/2) × a = a²/4

The shaded area is one-fourth of the square's area, which gives us the ratio 1:4.

Thus, the correct answer is C) 1:4.