Q14
Consider a right-angled triangle ABC, right angled at B. Two circles, each of radius r, are drawn inside the triangle in such a way that one of them touches AB and BC, while the other one touches AC and BC. The two circles also touch each other (see the image below).
If AB = 18 cm and BC = 24 cm, then find the value of r.
Answer: B
Set-up: place B at the origin, with BC along the x-axis (C = (24, 0)) and BA along the y-axis (A = (0, 18)). AC = 30.
- The circle touching AB and BC has centre (r, r).
- The second circle touches BC, so its centre is at height r. It touches the first circle, so the centres are 2r apart: centre (3r, r).
- Line AC: \( \frac{x}{24} + \frac{y}{18} = 1 \Rightarrow 3x + 4y = 72 \).
The second circle touches AC: the distance from (3r, r) to AC equals r.
\( \frac{72 - 3(3r) - 4r}{5} = r \Rightarrow 72 - 13r = 5r \Rightarrow r = 4 \)
Check: centres (4, 4) and (12, 4); distance from (12, 4) to AC = (72 − 36 − 16)/5 = 4.
Answer: 4 cm (Option 2)