What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale.)

maximum Aug 5, 2023

#1**+1 **

I think you have to find the areas of the right triangles surrounding the grey square and use that to find the ratio. You have to use the Pythagorean theorum to find the length of the slanted line and find the part which is the length of the grey square

jonathanldong Aug 5, 2023

#2**+1 **

The hypotenuse of the large right triangle at the left = sqrt (1^2 + (1+2)^2) = sqrt (10)

Looking at the small right triangle at the bottom left, notice that it is congruent to the small right triangle at the upper left

Call the longer leg of one of these triangles, x

The sin of the greater acute angle in these triangles = 3 / sqrt (10)

By the Law of Sines, we have

sin (90) /1 = 3/sqrt (10) / x

1 = [3 sqrt (10)] / x

x = 3/sqrt (10)

Call the smaller leg in either one of these triangles, y

The sin of the smaller acute angle in these triangles = 1/sqrt (10)

Again, by the Law of Sines, we have

sin (90) / 1 = 1/sqrt (10) / y

1 =[ 1/sqrt (10)] / y

y = 1/sqrt (10)

The length of the side of the square =

[The hypotenuse of the large right triangle on the left] - [ x ] - [ y ] =

sqrt (10 ) - 3/sqrt (10) - 1/sqrt (10)

sqrt (10) - [ 3 + 1] /sqrt (10)

(10 - 4) /sqrt (10) = 6/sqrt (10) = side of the shaded square

Shaded area = [ 6 / sqrt (10) ] ^2 = 36 / 10 = 3.6

Area of large square = 9

Ratio of areas = 3.6 / 9 = 36/90 = 4 / 10 = 2/ 5

CPhill Aug 5, 2023