Tangents from an outside point

Two tangents from one point

Take a point T outside a circle with centre O. Two tangents can be drawn from T. Let them touch the circle at P and Q.

The picture shows a pattern. The two tangent segments $TP$ and $TQ$ look equal. Let us prove it.

The two tangents are equal

Theorem. The lengths of the two tangents drawn from an outside point to a circle are equal.

Proof. Join OP, OQ and OT.

1. Right angles. A tangent is perpendicular to the radius at the point of contact. So $\angle OPT = 90^\circ$ and $\angle OQT = 90^\circ$.
2. Two right triangles. Look at $\triangle OPT$ and $\triangle OQT$. $OP = OQ$, because both are radii. $OT$ is common. Each has a right angle.
3. Congruent. The triangles are congruent by RHS: right angle, hypotenuse and one side.
4. Conclude. So $PT = QT$. $\blacksquare$

What else the proof tells us

The same congruence gives more.

• $\angle POT = \angle QOT$ and $\angle PTO = \angle QTO$. So OT bisects the angle $\angle PTQ$ between the tangents, and it also bisects $\angle POQ$.
• Triangle TPQ is isosceles, with $TP = TQ$. The line TO bisects its apex angle, so it is also the perpendicular bisector of the chord PQ.

So the centre of the circle lies on the line that bisects the angle between the two tangents.

The length of a tangent

In the right triangle OPT, the hypotenuse is OT. So

Example. A point T is $13$ cm from the centre of a circle of radius $5$ cm. Find the tangent from T.

1. The right triangle. $OP = 5$ cm and $OT = 13$ cm.
2. Pythagoras. $PT^2 = 13^2 - 5^2 = 169 - 25 = 144$.
3. Answer. $PT = 12$ cm, and $TQ = 12$ cm as well.

Example. The tangent from a point T is $24$ cm long, and T is $25$ cm from the centre. Find the radius.

1. The same triangle. $OP^2 = OT^2 - PT^2 = 625 - 576 = 49$.
2. Answer. The radius is $7$ cm.

The angle between two tangents

Look at the four-sided figure OPTQ. Two of its angles, at P and Q, are right angles. The four angles add up to $360^\circ$. So

The angle between the tangents and the angle between the two radii add up to $180^\circ$.

Example. The tangents TP and TQ meet at an angle of $70^\circ$. Find $\angle POQ$ and $\angle PTO$.

1. The angle at the centre. $\angle POQ = 180^\circ - 70^\circ = 110^\circ$.
2. OT bisects the angle at T. $\angle PTO = 70^\circ \div 2 = 35^\circ$.

Check. In the right triangle OPT, $\angle POT = 110^\circ \div 2 = 55^\circ$. Then $35^\circ + 55^\circ + 90^\circ = 180^\circ$.

Tangents at the ends of a diameter

Draw a diameter AB of a circle with centre O, and the tangents at A and B. Each tangent makes a right angle with the line AB. So the two tangents are parallel, because the angles on the same side add up to $180^\circ$. The distance between them is the length of the diameter.

The equal-tangent theorem holds only for two tangents from the same outside point. Tangents drawn from two different points are usually not equal.

Practise with new numbers

Try it in Python

The program finds the length of the tangent from a point T that is $13$ cm from the centre of a circle of radius $5$ cm. It then finds the two points of contact, checks that both are the same distance from T, and prints the angles.

Common mistakes

• Using OT as the tangent. OT is the hypotenuse. The tangent is PT, one leg of the triangle.
• Adding instead of subtracting. To find a tangent, subtract $r^2$ from $OT^2$. To find OT, add the squares.
• Saying all tangents to a circle are equal. Only the two from the same outside point are equal.
• Forgetting that the angles add up to $180^\circ$. $\angle POQ + \angle PTQ = 180^\circ$.

Remember these

• The two tangents from an outside point are equal: $TP = TQ$.
• $PT^2 = OT^2 - OP^2$.
• OT bisects the angle between the tangents.
• $\angle POQ + \angle PTQ = 180^\circ$.
• Tangents at the ends of a diameter are parallel.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Tangents TP and TQ are drawn from T to a circle. $TP = 9$ cm. What is $TQ$?

2. T is $10$ cm from the centre of a circle of radius $6$ cm. How long is the tangent from T?

3. The angle between two tangents is $60^\circ$. What angle do the radii to the points of contact make at the centre?

4. In the last question, what is $\angle PTO$?

5. The tangent from T is $8$ cm long and T is $17$ cm from the centre. Find the radius.