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The length of the tangent to a circle from a point 17cm from its center is 8cm. Find the radius of the circle.

Last updated date: 25th Jun 2024
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Hint: We know that tangent is always perpendicular to the radius made from that point, using this point we’ll use the property of right-angled triangle so formed.
After using the property we’ll get an equation and solving that equation we’ll get the value of the radius of the circle.

Complete step-by-step answer:
Given data: \[Length{\text{ }}of{\text{ }}tangent = 8cm\]
The distance of the point from centre=17cm
Let the centre of the circle be ‘O’ and the point from which the tangent is made be ‘R’, the point tangent touches the circle be ‘P’.
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We know that tangents are always perpendicular to the radius made from that point.
Therefore, \[OP \bot PR\] or $\angle P = {90^ \circ }$
Using Pythagoras theorem
i.e., in a right-angled triangle ABC with right angle at A, $B{C^2} = A{B^2} + C{A^2}$
Therefore, Using Pythagoras theorem in triangle OPR where $\angle P = {90^ \circ }$
$ \Rightarrow O{R^2} = P{R^2} + P{O^2}$
Substituting the value of OR and PR
$ \Rightarrow {17^2} = {8^2} + P{O^2}$
On Separating the unknown term,
$ \Rightarrow P{O^2} = {17^2} - {8^2}$
On squaring we get,
$ \Rightarrow P{O^2} = 289 - 64$
$ \Rightarrow P{O^2} = 225$
Taking positive square root on both sides, we get,
$\therefore PO = 15cm$
Therefore, the radius of the circle is 15 cm.

Note: Here we have given the tangent of the circle, so let us discuss some properties related to the tangent of a circle
1.A tangent of a circle always touches the circle at a single point.
2.Tangent is always perpendicular to the radius made at the point of tangency.
3.The length of two tangents drawn to a single point to a circle is always equal.