Question

Make an isosceles triangle with a radius of 4 cm and measure the length of one side of it.​

Answer 1

Answer:

What is the length of the sides of a triangle when the radius of incircle is 4 cm?

GIVEN: Triangle ABC. AB, BC , AC are tangents at points P, Q, R to incircle with centre ‘O'

As we know tangent segments from an exterior point to a circle are equal in length…

So, AP = AR = Z

BP = BQ = X

CQ = CR = Y

Radius of the incircle = 4 cm.

CALCULATION: Method is: We find out the area of triangle ABC in 2 different ways. Then equate them….

Ar( triangleABC) =√ {s(s-a)(s-b)(s-c)} ( Heron's formula)

S = semi perimeter =( 2x +2y +2z )/2 = x+y+z

& a, b, c are sides of the triangle

Ar( tri ABC)= √{(x+y+x)xyz}………….(1)

Also ar(tri ABC) = ar(tri OBC)+ar(tri OCA) + ar(tri(OBA)

ar( triOBC) = 1/2*(x+y)*4=2(x+y)

ar(tri OCA) = 1/2*(y+z)*4 = 2(y+z)

ar(tri OBA) = 1/2*(x+z)*4= 2(x+z)

Hence ar(tri ABC) = 2(x+y) + 2(y+z) + 2(x+z)

ar( tri ABC) = 4(x+y+z)……..(2)

Eq(1) = Eq(2)

√{(x+y+z)(x)(y)(z)} = 4(x+y+z)

=> (x+y+z)(x)(y)(z) = 16(x+y+z)²

=> xyz = 16(x+y+z)……….ANS

Here, if any 2 variables are known, we can calculate all the 3 sides of the triangle. Length of the sides of the triangle depend on the ratio of x:y or y:z or x:z