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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 lengths 8 cm and 6 cm respectively (see figure). Find the sides AB and AC.

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AC = 13 cm, AB = 15 cm

AC = 14 cm, AB = 15 cm

AC = 13 cm, AB = 14 cm

AC = 14 cm, AB = 16 cm

answer is A.

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Detailed Solution

Given that a triangle ABC is drawn to circumscribe a circle of radius 4 cm.
Question Image

The circle of radius r=4cm.
Semi-perimeter of triangle ABC by using the Heron's formula,

s = Sum of sides of trangle 2 . ⇒ s = 14 + (x + 6) + (x + 8) 2 ⇒ s = 2 x + 28 2 ⇒ s = x + 14

The area of a triangle can be evaluated as,

A r e a = s (s − a) (s − b) (s − c) ⇒ A = (x + 14) (x + 14 − 14) (x + 14 − x − 6) (x + 14 − x − 8)

⇒ A = (x + 14) x (8) (6) ⇒ A = 48 x (x + 14) ---(1)

From the figure,
Area of a triangle =

A (Δ A C O) + A (Δ A B O) + Δ B C O

⇒ A = 12 (4) (14) + 12 (4) (6 + x) + 12 (4) (8 + x) ⇒ A = 28 + 12 + 2 x + 16 + 2 x ⇒ A = 4 x + 56 ---(2)

From (1) and (2) we get,

⇒ 48 x (x + 14) = 4 x + 56 ⇒ 48 x (x + 14) = 16 (x + 14) 2 ⇒ 3 x = (x + 14) ⇒ 2 x = 14

⇒2x=14
⇒x=7
The length of side AC is 7+6=13.
Length of side A B is 7+8=15.
Hence, the length of side AC is

13 cm

and length of side AB is

15 cm .

Therefore, the correct answer is option (1).

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