Find the area of the right triangle

Question

The circumradius of the triangle is 10 cm and the inradius is $4$cm.

A.) $28$

B.) $56$

C.) $96$

D.) $192$

Answer 1

Assuming triangle is right amgle triangle.

The circumradius is bisector of the hypotenuse. So hypotenuse = 20.

Let other two sides x and y.

x + y = 2(Inradius + Circumradius)

x + y = 2(4 + 10)

x + y = 28

x = 28 - y ..........(1)

Sum of all three sides = 20 + 28 = 48

Also Product of other two sides except hypotenuse is equal to the product of sum of all sides × in radius

xy = 48 × 4

From equation (1) putting the value of x,

(28 - y)y = 192

$y^2 - 28y + 192 = 0$

$y^2 - 16y - 12y + 192 = 0$

y = 16 or 12

And x = 12 or 16

Area = $\frac12 × 12 × 16 = 96$