Questions & Answers

Question

Answers

Answer
Verified

It is given that; a tangent \[PQ\] at a point \[P\] of a circle of radius \[5\]cm meets a line through the centre \[O\] at a point \[Q\]so that \[OQ = 13\]cm.

We have to find the length of \[PQ\].

We know that the tangent of a circle is perpendicular with its radius at the point of tangent.

So, here the radius \[OP\] is perpendicular to the tangent \[PQ\]. So, \[\angle OPQ = {90^ \circ }\].

Therefore, \[\Delta OPQ\] is a right-angle triangle whose \[\angle OPQ = {90^ \circ }\]and \[OQ = 13\]cm

So, we can apply Pythagoras theorem.

We have,

\[O{Q^2} = O{P^2} + P{Q^2}\]

Substitute the values we get,

\[{13^2} = {5^2} + P{Q^2}\]

Simplifying we get,

\[P{Q^2} = 169 - 25\]

Simplifying again we get,

\[PQ = \sqrt {144} = 12\] (we will take the positive value)

Hence, the length of \[PQ\] is \[12\] cm.

Here, \[PQ\] gives another value that is \[ - 12\]. But the length of any object cannot be negative. So, we ignore negative values for the length of \[PQ\].

Tangent to a circle is a line that touches the circle at one point, which is known as point of tangent. At the point of tangent, the tangent of a circle is always perpendicular to the radius.