What is AB+BC equal to?
Consider the following for the next three (03) items that follow: ABC is a right-angled triangle with \angle ABC=90^{\circ}. The centre of the incircle of the given triangle is at O, whose radius is 2 cm. Two more circles with centres at O_{1} and O_{2}, touch this circle and the two sides as shown in the figure given below. Further, MA:MC=2:3.
- A. 10 cm
- B. 12 cm
- C. 13 cm
- D. 14 cm ✓
Correct Answer: D. 14 cm
Explanation
Let the segments of the hypotenuse be AM = 2x and CM = 3x. Due to tangent properties, the sides are AB = 2x + 2 and BC = 3x + 2. Using Pythagoras: (2x+2)^2 + (3x+2)^2 = (5x)^2. This simplifies to 13x^2 + 20x + 8 = 25x^2 \implies 12x^2 - 20x - 8 = 0 \implies 3x^2 - 5x - 2 = 0. Factoring gives (3x+1)(x-2) = 0, so x=2. Therefore, AB = 6 and BC = 8. Their sum is 14 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...