The given figure shows a circle with centre O and radius 4 cm circumscribed by ΔABC.

Question

The given figure shows a circle with centre O and radius 4 cm circumscribed by \(\Delta\)ABC. BC touches the circle at D such that BD = 6 cm, DC = 10 cm. Find the length of AE. 

Answer 1

Let AE = x

CD = CE = 10

Tangents from same external point

Using Herons Formula 

ar \((\Delta ABC) = \sqrt{(x+16)\times x\times10 \times 6}\) 

also ar\((\Delta ABC) = ar(AOB) + ar(BOC) + ar(\Delta AOC)\) 

= \(\frac{1}{2} \times 4 \times (x+6) + \frac{1}{2} \times 4 \times (16) + \frac{1}{2} \times 4 \times (x+10)\)  

= \(\frac{1}{2} \times 4 [ x+ 6 +16+x+10]\)  

= 2(32 +2x) = 4 (x+16) 

\(4(x+16) = \sqrt{(x+16)\times 10\times6 \times x}\)    [using Heron's formula]

Squaring both sides we get

16(x + 16)² = x(x + 16) × 10 × 6

4(x + 16)= 15x

4x + 64 = 15x

11x = 64 

\(x = \frac{64}{11}\)