Find the Sum of the Area of Two Squares in a Semicircle

Geometry math problem: Two squares are perfectly inscribed inside a semicircle, then find the sum of the area of squares when the diameter of the semicircle is given

Find the sum of the area of two squares (blue region) shown below, here diameter of the semicircle is 20 cm

Solution to this geometry math problem

Let’s consider sides of the squares are x and y, also name the figure as shown below

Now connect BF, BD and DF then we get a triangle BDF

From the figure

BD and BF diagonals of the square, so

∠DBF = 45 + 45 = 90°

BF = y√2

BD = x√2

Apply Pythagoras theorem in the triangle BDF

DF² = 2y² + 2x²

Now take reflection of this figure then we get arc DF = 90° (KF intersecting ∠ AKC)

Then consider  ends of chord DF connect with centre then we get a right triangle

From figure DF² = 10² + 10² = 200 

We also know DF² = 2x² + 2y² 

Then 2x² + 2y² = 200

⇒ x² + y² = 100

That is, sum of the area of the squares = 100 cm²