Question

In the given figure, ΔABC circumscribes a circle of radius of 4 cm. If AD = 7 cm, BD = 8 cm and area (ΔАBC) = 84 cm², find the lengths of BC and AC.

Answer 1

Given:

Triangle ABC circumscribes a circle of radius r = 4 cm.

The circle is tangent to AB at D with AD = 7 cm and BD = 8 cm, so AB = 7 + 8 = 15 cm.

Area (ΔABC) = 84 cm².

Step-wise calculation:

1. Let side lengths be a = BC, b = CA, c = AB.

Semiperimeter `s = (a + b + c)/2`

2. For a triangle with incircle radius r, Area = r × s. 

So, `s = "Area"/r` 

= `84/4`

= 21

3. c = AB = 15

Therefore a + b + 15 = 2s = 42

⇒ a + b = 27

4. Tangent-length property: the tangents from a vertex to the incircle are equal.

Thus, AD = s – a = 7

BD = s – b = 8

Subtracting gives (s – b) – (s – a)

= a – b 

= 8 – 7

= 1

So, a = b + 1.

5. Solve the system:

a + b = 27

a = b + 1

Substitute: (b + 1) + b = 27 

⇒ 2b + 1 = 27

⇒ 2b = 26

⇒ b = 13

Then a = b + 1 = 14.

BC = a = 14 cm.

AC = b = 13 cm.