The radius of the in-circle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and 8 cm. Determine the other two sides of the triangle.
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See the diagram.
Tangents from B onto the circle are equal. So BF=8cm and AE=6cm.
Triangles ODB and OFB are congruent. Similarly triangles OAD and OEA.Similarly triangles OCE and OCF.
Angle DBO and angle OBF are equal and are equal to 1/2*angle B.Angle OAD and angle OAE are equal and are equal to 1/2 * angle A.Angles OCE and angle OCF are equal and are equal to 1/2 * angle C.
Hence,2 * tan^-1 4/8 + 2 * tan^-1 4/6 + 2 * tan^-2 4/x = 180 deg.Tan^-2 1/2 + tan^-1 2/3 = 90 deg - tan^-1 4/xTangent of both sides:(1/2 + 2/3) / (1- 1/2 * 2/3) = cot [tan^-1 4/x] = x/4
So x = 4*7/6 / (2/3) = 7 cm
So the other sides are : AC = 6+7 = 13 cmand BC = 8+7 = 15 cm.
Tangents from B onto the circle are equal. So BF=8cm and AE=6cm.
Triangles ODB and OFB are congruent. Similarly triangles OAD and OEA.Similarly triangles OCE and OCF.
Angle DBO and angle OBF are equal and are equal to 1/2*angle B.Angle OAD and angle OAE are equal and are equal to 1/2 * angle A.Angles OCE and angle OCF are equal and are equal to 1/2 * angle C.
Hence,2 * tan^-1 4/8 + 2 * tan^-1 4/6 + 2 * tan^-2 4/x = 180 deg.Tan^-2 1/2 + tan^-1 2/3 = 90 deg - tan^-1 4/xTangent of both sides:(1/2 + 2/3) / (1- 1/2 * 2/3) = cot [tan^-1 4/x] = x/4
So x = 4*7/6 / (2/3) = 7 cm
So the other sides are : AC = 6+7 = 13 cmand BC = 8+7 = 15 cm.
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