Tangent to circle properties — length, angle, power of a point

Question

What are the key properties of a tangent to a circle, and how do we use them to solve problems involving tangent lengths and angles?

Solution — Step by Step

A tangent to a circle at any point is perpendicular to the radius drawn to the point of contact.

OT \perp PT

where $O$ = centre, $T$ = point of tangency, $P$ = external point.

This creates a right angle at $T$ , which is the foundation of almost every tangent problem. When you see a tangent, immediately draw the radius to the point of contact and mark the 90-degree angle.

From any external point $P$ , the two tangent segments drawn to a circle are equal in length:

PA = PB

where $A$ and $B$ are the points of tangency.

Also, the line $OP$ bisects the angle $\angle APB$ and bisects the chord $AB$ perpendicularly.

This property is the most frequently tested in CBSE Class 10.

From external point $P$ at distance $d$ from centre $O$ of a circle of radius $r$ :

\text{Tangent length} = PT = \sqrt{d^2 - r^2}

This comes directly from the right triangle $OTP$ where $OT = r$ , $OP = d$ , and angle $OTP = 90°$ .

Example: Point $P$ is 13 cm from the centre of a circle of radius 5 cm. Tangent length = $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm.

When a circle is inscribed in a triangle (incircle), tangent lengths from each vertex are equal. If the tangent lengths from vertices $A$ , $B$ , $C$ are $x$ , $y$ , $z$ respectively:

a = y + z, \quad b = x + z, \quad c = x + y

where $a$ , $b$ , $c$ are the sides opposite to vertices $A$ , $B$ , $C$ .

Solving: $x = s - a$ , $y = s - b$ , $z = s - c$ where $s = (a + b + c)/2$ is the semi-perimeter.

flowchart TD
    A["Tangent Problem"] --> B{"What is given?"}
    B -->|"External point and circle"| C["Find tangent length: sqrt of d2 minus r2"]
    B -->|"Two tangents from a point"| D["PA = PB, use equal tangent lengths"]
    B -->|"Circle inscribed in triangle"| E["Tangent lengths = s minus opposite side"]
    B -->|"Angle between tangent and radius"| F["Always 90 degrees at point of contact"]
    C --> G["Use right triangle OTP"]
    D --> H["OP bisects angle APB"]

Why This Works

The perpendicularity of tangent and radius follows from the definition: a tangent touches the circle at exactly one point. If the tangent were not perpendicular to the radius, it would intersect the circle at two points (creating a secant), which contradicts the definition.

Equal tangent lengths from an external point follow from congruent triangles: triangles $OPA$ and $OPB$ are congruent by RHS (they share hypotenuse $OP$ , $OA = OB$ as radii, and both have right angles at the tangent points).

Alternative Method

For problems involving the angle between two tangents from an external point, use:

\angle APB = 2\sin^{-1}\left(\frac{r}{d}\right) = 2\arctan\left(\frac{r}{PT}\right)

Or equivalently, if $\angle APB = \theta$ , then $\angle AOB = 180° - \theta$ (they are supplementary in the quadrilateral $OAPB$ ).

Common Mistake

In CBSE problems with a circle inscribed in a quadrilateral, students forget the key property: the sum of opposite sides of a circumscribed quadrilateral are equal ( $AB + CD = BC + DA$ ). This follows directly from equal tangent lengths. Many 4-mark CBSE questions give three sides and ask for the fourth — the answer is one line using this property, but students draw elaborate constructions because they do not know it.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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