Q. 23

# Let us write by calculating the length of radius of incircle and circumcircle of a triangle of which sides are 20 cm, 15 cm, 25 cm. Let us calculate the area of the regions bounded by incricle and circum-circle.

Answer :

Let BCD is a triangle of sides BC, BD and DC equal to 20 cm, 15 cm, 25 cm. AB, AD and AC are the internal bisectors of respective angles of the triangle. The three internal bisectors meet at A. Perpendiculars drawn from A on sides BC, CD and DB are AG, AE and AF respectively.

AG = AE = AF

Legth of inner radius of triangle = AG

Let AG be r units.

The given triangle forms a right angled triangle with ∠ B = 90°

Using Pythagoras theorem,

BE^{2} +DE^{2} = BD^{2}

⇒ BE^{2} + 12.5^{2} = 15^{2}

⇒ BE^{2} = 68.75

⇒ BE = 8.3 cm

The centroid of triangle is at A and lies on height BE.

⇒ AE = 2.767 cm

r = 2.767 cm

∵ Area of a circle = πr^{2}

⇒ Area of incircle = 24 cm^{2}

∵The given triangle forms a right angled triangle

∴ BC will act as the diameter of the circumcircle centered at O.

Diameter = 25 cm

⇒ Radius = 12.5 cm

∵ Area of a circle = πr^{2}

⇒ Area of the circumcircle

⇒ Area of circumcircle = 491 cm^{2}

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