In the diagram given below, the circles having centres at $P,Q$and $R$ touch each other external

Question

In the diagram given below, the circles having centres at $P,Q$and $R$ touch each other externally. Given that the radius of the three circles are $3$ cm, $x$ cm and $\left( 3x+2 \right)$ cm respectively, and $\angle PQR=90{}^\circ $, form an equation in $x$ and show that it reduces to $2{{x}^{2}}-2x-3=0$. Solve the equation $2{{x}^{2}}-2x-3=0$, giving your answers correct to $2$ decimal places. Hence, calculate the perimeter of the triangle $PQR$.

Accepted Answer

(a) 2x^2 - 2x - 3 = 0; (b) x = -0.82 or 1.82; (c) 24.6 cm.

Geometry Problem Involving Circles and Quadratic Equations

Geometry Problem Involving Circles and Quadratic Equations

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