Two circles touch externally. The sum of their areas is 89 square cm and the distance between their centres is 13 cm. What is the difference in their radii?

Explanation

Let the radii be r_1 and r_2. Given r_1 + r_2 = 13. Assuming the sum of areas is 89\pi (typographical error in the original paper dropping \pi), we have \pi r_1^2 + \pi r_2^2 = 89\pi \implies r_1^2 + r_2^2 = 89. Using the identity (r_1+r_2)^2 - 2r_1 r_2 = 89 \implies 169 - 2r_1 r_2 = 89 \implies 2r_1 r_2 = 80. Then (r_1-r_2)^2 = r_1^2 + r_2^2 - 2r_1 r_2 = 89 - 80 = 9 \implies r_1-r_2 = 3\text{ cm}.

Related questions on Geometry

• In a triangle ABC, \angle A = 30^\circ, AB = 7 cm and AC = 12 cm. What is the area of the triangle ABC?
• ABC is a triangle right angled at B. D is a point on AC such that BD is perpendicular to AC. If AB = p and BC = \sqrt{3}p, then what is BD...
• The difference between an interior angle and an exterior angle of a regular polygon is 120°. What is the number of sides of the polygon?
• An angle q is exactly one-fourth of its complementary angle. What is the value of angle q?
• The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the area of the triangle formed by joining the mid-points of the sides of the tr...