In Fig. OQ:PQ = 3:4 and perimeter of ΔPOQ = 60 cm Determine PQ, OR and OP.

Question

In Fig. OQ:PQ = 3:4 and perimeter of \(\Delta POQ=60\) cm. Determine PQ, OR and OP.

Answer 1

Given that OQ: PQ = 3:4

Let ratio coefficient =x, so

OQ = 3x and PQ = 4x

We know that a tangent to a circle is perpendicular to the radius at the point of tangency

So

\(\angle OQP=90^\circ\)

Then applying Pythagoras theorem in triangle POQ

\(OP^2=OQ^2+PQ^2\)

\(OP^2=(3x)^2+(4x)^2\)

\(OP^2=9x^2+16x^2\)

\(OP^2=25x^2\)

\(OP=5x\)

Perimeter of a triangle POQ is = 60 cm, so

\(3x+4x+5x=60\)

\(12x=60\)

\(x=5\)

So,

\(OQ=3x=15\,cm\)

\(PQ=4x=20\,cm\)

\(OP=5x=25\,cm\)

\(QR=2(OQ)=2\times15=30\,cm\)