Given a semicircle with diameter 5, three squares are constructed as shown below. Find the sum of the areas of the three squares.
+128406 Let the eqaution of the circle be x^2 + y^2 = 2.5^2 ⇒ x^2 + y^2 = 6.25
Call the side of the purple square, a
Then, when y = a , we can find x as sqrt ( 6.25 - a^2 )
And using symmetry, the distance from P to this point = 2 sqrt (6.25 - a^2)
Using the secant-tangent theorem, we have that
a^2 = [ (1/2)a ] [( (1/2)a + 2sqrt ( 6.25 - a^2 ]
a^2= (1/4)a^2 + a sqrt (6.25 -a^2)
(3/4)a^2 = a sqrt ( 6.25 -a^2) divide out a
(3/4)a = sqrt (6.25 - a^2) square both sides
(9/16)a^2 = 6.25 - a^2
(9/16)a^2 = 25/4 - a^2
(25/16)a^2 = 100/16
25a^2 =100
a^2 = 100/25 = 4
a = 2
So....the area of the pink square = a^2 = 2^2 = 4
And the area of the orange square is [ (1/2) a]^2 = [(1/2)(2)]^2 = 1
And the area of the aqua suare is ( 5 -2)^2 =3^2 = 9
So....the total area is 4 + 1 + 9 = 14
