Question

Simple Math Puzzle on triangles.

ABCD is a square. Green circle's radius = 2 cm. Blue circle has a radius = 4 cm.
Find the length "a" of the edge of the square.

Use the formulas related to triangles.

Answer 1

Heya User,

Clearly.
--> PF = KO = KF = 4cm
--> BP = BT ; CK = CT --> [ Tangents to a circle from common point ]

=> Perimeter [ BFC ] = BP + PF + KF + KC + BC
--> Perimeter [ BFC ] = ( BT + CT ) + PF + KF + BC
--> Perimeter [ BFC ] = a + a + 4 + 4
--> Perimeter [ BFC ] = 2a + 8; --------> (i)

Also,
==> AREA [ BFC ] =inradius[ Semi-Perimeter [ BFC ] ]--------> (ii)

Now, Area [ EFGH ] = 4² = 16; ---------> (iii)

Observe :-> ΔBFC, AEB, AHD, DGC are same !!

---> .'. 4(ii) + (iii) ---> gives :->

----> 16[ 2a + 8 ]/2 + 16 = a²
==> 16a + 64 + 16 = a²
==> a² - 16a - 80 = 0
==> a = 20, -4

Hence, a = 20 is the soln.

Answer 2

Nice question.
Answer explained below.
Comment below if any doubt occurs to you.