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Question
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of 2 ABC 21cm then find the lengths of sides AB and AC.
Solution
Construction: Join OA, OB, OC, OE ⊥ AB at E and OF ⊥ AC at F
We know that tangent segments to a circle from the same external point are congruent
Now, we have
AE = AF,BD = BE = 4cmand CD = CF = 3cm
Now,
Area (ΔABC) = Area(ΔBOC)+ Area(ΔAOB) + Area(ΔAOC)
`⇒21 = 1/2 xx BC xx OD +1/2 xx AB xx OE +1/2 xx AC xx OF`
`⇒ 42 = 7 xx 2 + (4+x) xx 2 +(3+ x) xx2`
⇒ 21 = 7 + 4 + x + 3 + x
⇒ 21=14+2x
⇒ 2x =7
⇒ x= 3.5 cm
∴ AB = 4 + 3.5 = 7.5 cm and AC = 3 +3.5 = 6.5 cm
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