A circle is defined as the locus of a point moving in a plane such that its distance from a fixed point always remains constant. This fixed point is called the centre of the circle. The fixed distance is called the radius of the circle and the locus or the boundary of the circle is called the circumference of the circle.

Define a tangent to a circle.

In geometry, a tangent to a circle is a straight line that only touches or intersects the circle at one point. A tangent is a line that never passes through the centre of the circle. A circle with a point X is depicted in the diagram below. A tangent C has been drawn from X . This is an example of a circle tangent.

P Q and P T are tangents to a circle with centre O and radius 5 cm . If P Q = 12 cm , then prove that the perimeter of the quadrilateral P Q O T is 34 cm .

Given: P Q and P T are tangents to a circle and radius is 5 cm and P Q = 12 cm . Here, P Q and P T are the tangents drawn from an external point P . We know that, If two tangent segments are drawn from the same external point, then the segments are equal. ∴ P Q = P T = 12 cm and O Q = O T = 5 cm [Radii of the circle] So, Perimeter of P Q O T = Sum of all sides = P Q + Q O + O T + P T = 12 + 5 + 5 + 12 = 34 cm Hence, proved.

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Let A P and A Q are two tangents from an external point A to a circle with centre O . To Prove: A P = A Q The figure from the question can be drawn as follows. Join O P , O Q and O A . We know that, a tangent at any point on a circle is perpendicular to the radius through the point of contact. Here, A P is a tangent and O P is the radius of the circle through P . ∴ O P ⊥ A P Similarly, O Q ⊥ A Q ⇒ ∠ O P A = ∠ O Q A = 90 ° . . . ( i ) In Δ O P A and Δ O Q A , we have O P = O Q [ radii of a same circle ] ∠ O P A = ∠ O Q A = 90 ° [from Eq. (i)] O A = O A [common sides] So, Δ O P A ≅ Δ O Q A [ by R.H.S Congruence rule] ∴ A P = A Q [by CPCT] Hence, proved.