Question

PQ is a chord of length 8 cm to a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length of the tangent TP. [4 marks]

Answer 1

Let TR = y . Since, OT is perpendicular bisector of PQ.

PR = QR = 4 cm

In Triangle ORP,
$O{P}^2=O{R}^2+P{R}^2\Rightarrow O{R}^2=O{P}^2-P{R}^2\Rightarrow O{R}^2=5^2-4^2=25-16=9\Rightarrow OR=3cm[0.5\mathrm{marks}]\Rightarrow OT=OR+RT=3+y------\left(1\right)[0.5\mathrm{marks}]In\Delta PRTT{P}^2=T{R}^2+P{R}^2------\left(2\right)In\Delta OPT,O{T}^2=T{P}^2+O{P}^2[0.5\mathrm{marks}]O{T}^2=(T{R}^2+P{R}^2)+O{P}^2[0.5\mathrm{marks}]{(3+y)}^2=y^2+4^2+5^{2}9+6y+y^2=y^2+16+256y=41-9\mathrm{We}\mathrm{get}y=16/3cm\left[1\mathrm{mark}\right]\mathrm{Also},T{P}^2=T{R}^2+P{R}^2\Rightarrow T{P}^2={\left(\frac{16}{3}\right)}^2+4^2\Rightarrow TP=\frac{20}{3}cm\left[1\mathrm{mark}\right]$