Dare2solve | Area of Quarter Circle drawn by vertex of two squares

This question was uploaded on 11/08/21 on social media accounts.

Two squares are given on the same baseline. One with side length 2 and the other is unknown. A quarter circle is drawn with the center on the case line of the large square. One vertex of the quarter circle is on the vertex of one square and another vertex is on the vertex of the other square. The quarter circle is passing through another vertex of the square. Find the area of Quarter circle.

Step 1:

Give names to all Vertex.

Join Center of quarter circle and vertex of square.

Join Center of quarter circle and Mid Point of Top side of square

Let side of large square is 'x'.

Step 2:

Observe triangles ABO and OEF:

AO = OF = Radius of Quarter circle

Angle ABO = Angle OEF = 90

Angle BOA = Angle EFO

By this Both triangles are congruent.

⇒ AB = OE = x

CE = 2

⇒ OC = 2 - x

Step 3:

Observe triangles AOM and DOM:

AO = OD = Radius of Quarter circle

Angle AMO = Angle DMO = 90

OM is common side

By this Both triangles are congruent.

⇒ AM = MD

⇒ BO = OC

⇒ 2 = 2 - x

⇒ x = 4

Step 4:

Observe triangle ABO:

AB = 4

BO = 2

B is 90^o  

⇒ AB² + BO² = AO² 

⇒ AO² = 16 + 4 = 20

⇒ AO² = √20

Step 5:

Area of quarter circle = π(√20)² /4 = 5π

Image Solution:

Solution From Hossein Rezvani:

Puzzle related to Geometry, Area, Square, Circle.