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A triangle has side lengts 2,3,4. a tangent is drawn to the incircle parallel to side 2cm cutting the other two sides at M and N .Find the lenght MN.
Asked by sandy72434 | 07 Jul, 2019, 01:52: PM
Triangle ABC with a = 2, b = 3 and c = 4

let r be the radius of the in circle and a, b and c are the sides of the triangle ABC.

and XY || AC

area of the triangle ABC = 1/2 × bh and in terms of in circle radius.

area of triangle ABC = 1/2 × r × (a + b + c)

therefore

1/2 bh = 1/2 × r × (a + b + c)

r/h = b/(a + b + c) ....(i)

Triangles BXY and triangle BAC are similar triangles.

Therefore

MQ/AC = (h - 2r)/h

= 1 - 2r/h

= 1 - 2b/(a + b + c)

= (a - b + c)/a + b + c) ....(ii)

Substitute AC = b = 2 , a = 3 and c = 4 in eq(ii):

MQ/2 = (3 - 2 + 4)/(2 + 3 + 4) = 5/9

MQ = 10/9

Answered by Sneha shidid | 11 Jul, 2019, 01:31: PM

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