Question

3E. Given square ABCD, with point P on side AB , point Q on side BC, point R on side CD, and point S on side DA as shown. Additionally, P is 2/3 of the way from A to B, Q is 2/3 of the way from B to C, R is 2/3 of the way from C to D, and S is 2/3 of the way from D to A. Compute the ratio of the area of square PQRS to the area of square ABCD. Write your answer as a fraction in lowest terms.

Answer 1

The length of the sides of the square PQRS is a fraction of the

length of the sides of square ABCD.

Correct response:

• The ratio, of area of square PQRS to the area of square ABCD is; 5:9

Which is the method used to find the ratio of the areas?

Let x represent the side length of the square, ABCD we have;

Area of the square, ABCD is A = x²

Side length, s, of the inscribed square, PQRS, is given as follows;

• $s = \displaystyle \sqrt{ \left(\frac{2}{3} \cdot x \right)^2 + \left(\dfrac{1}{2} \cdot x\right)^2 } = \frac{\sqrt{5} }{3} \cdot x$

$Area \ of \ the \ inscribed \ square\ = s^2 = \left(\dfrac{\sqrt{3} }{3} \cdot x \right)^2 = \mathbf{\dfrac{5}{9} \cdot x^2}$

$Ratio \ of \ area \ of \ square \ PQRS \ to \ square \ ABCD = \dfrac{\dfrac{5}{9} \cdot x^2 }{x^2} = \mathbf{ \dfrac{5}{9}}$

• The ratio of the area of the square PQRS to the square ABCD is 5:9

Learn more about finding the area of geometric shapes here:

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