A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of the lengths 8 cm and 6 cm respectively. Find the sides AB and AC.

Question

A triangle ABC is drawn to circumscribe a circle of radius 4 cm  such that the segments BD and DC into which BC is divided by the point of contact D are of the lengths 8 cm and 6 cm respectively. Find the sides AB and AC.

Filo · Accepted Answer

Let the circle touch the side AB and AC of the triangle at point E and F.

In

△ A BC

,

CF = C D = 6

cm ....Tangents from an external point to a circle are equal

BE = B D = 8

cm ....Tangents from an external point to a circle are equal

A E = A F = y

....Tangents from an external point to a circle are equal

Now,

A B = A E + EB

....

[∵ A - E - B]

∴ A B = (x + 8)

cm

Similarly,

BC = B D + D C = 14

cm

and

C A = CF + F A = (6 + x)

cm

Perimeter of

(Δ A BC) = A B + BC + A C

= x + 8 + 14 + x + 6

= 2 x + 28

s = \frac{2 x + 28}{2} = x + 14

By Heron's formula,

Area of

Δ A BC = s (s - a) (s - b) (s - c)

A (Δ A BC) = (x + 14) [(x + 14) - 14] [(x + 14) - (6 + x)] [(x + 14) - (8 + x)]

= (x + 14) x \times 8 \times 6

= 4 3 (x^{2} + 14 x)

Area of

Δ OBC = \frac{1}{2} \times O D \times BC = \frac{1}{2} \times 4 \times 14 = 28

Area of

Δ OC A = \frac{1}{2} \times OF \times A C = \frac{1}{2} \times 4 (x + 6) = 2 x + 12

Area of

Δ O A B = \frac{1}{2} \times OE \times A B = \frac{1}{2} \times 4 (x + 8) = 2 x + 16

A (Δ A BC) = A (Δ OBC) + A (Δ OC A) + A (Δ O A B)

∴ 4 3 (x^{2} + 14 x) = 28 + 2 x + 12 + 2 x + 16

\Rightarrow 3 (x^{2} + 14 x) = x + 14

Squaring on both sides we get,

\Rightarrow 3 (x^{2} + 14 x) = (x + 14)^{2}

\Rightarrow 3 x = x + 14

\Rightarrow 2 x = 14

\Rightarrow x = 7

So,

A B = x + 8 = 15

cm.

and

A C = x + 6 = 13

cm.

Question Text	A triangle $A BC$ is drawn to circumscribe a circle of radius $4 c m$ such that the segments $B D$ and $D C$ into which $BC$ is divided by the point of contact $D$ are of the lengths $8 c m$ and $6 c m$ respectively. Find the sides $A B$ and $A C$ .
Answer Type	Text solution:1
Upvotes	150

A triangle $A BC$ is drawn to circumscribe a circle of radius $4 c m$ such that the segments $B D$ and $D C$ into which $BC$ is divided by the point of contact $D$ are of the lengths $8 c m$ and $6 c m$ respectively. Find the sides $A B$ and $A C$ .

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