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.

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MrRoyal MrRoyal · Answer 1 · 2022-01-14T15:18:39+01:00

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

Assume the measure of the side lengths of the square ABCD is 1, then the area of the square ABCD is:

[tex]A_1 = 1 \times 1[/tex]

[tex]A_1 = 1[/tex]

See attachment for the diagram that represents the relationship between both squares.

The measure of length PQ is then calculated using the following Pythagoras theorem:

[tex]PQ^2 = (\frac 13)^2 + (\frac 23)^2[/tex]

Evaluate the squares

[tex]PQ^2 =\frac 19 + \frac 49[/tex]

Add the fractions

[tex]PQ^2 =\frac 59[/tex]

The above represents the area of the square PQRS.

i.e.

[tex]A_2 =\frac 59[/tex]

So, the ratio of the area of PQRS to ABCD is:

[tex]Ratio = \frac{A_2}{A_1}[/tex]

This gives

[tex]Ratio = \frac{5/9}{1}[/tex]

[tex]Ratio = \frac{5}{9}[/tex]

Hence, the ratio of the area of square PQRS to the area of square ABCD is 5/9