Question

Find the lengths of the sides of a right triangle if R = 15 cm and r = 6 cm, where R and r are the radii of the circumscribed and the inscribed circle respectively.

Answer 1

Let, a,b be the legs of the triangle and c be its hypotenuse

For a right triangle, the hypotenuse is the diameter of the circum circle. Hence $c=2R=30$

Also, we know that the sum up legs of a right angled triangle is double the sum of the radii of the circum circle and the in circle.

Thus $a+b=2(R+r)=42⟹b=42-a$

By pythogoras theorem, we have, $a^2+b^2=c^2=900$

$⟹a^2+(42-a{)}^2=900$

$⟹a^2-42a+432=0$

Solving this we get, a=24,18

Thus the sides of the right triangle are $18,24$ and $30$.