SPI Mathematics · Circles

Tangent and Radius

A tangent touches a circle at one point, and the radius drawn to that point meets the tangent at a right angle.

Why this matters

Fourteen questions in the bank, thirteen of them EASY: what a tangent is, the gap between parallel tangents, and the length of a tangent from a right triangle. Almost every one is a single use of Pythagoras, so the mark is lost only by putting the right angle at the wrong corner.

Concept 1 of 3: What a tangent is and how many there are

A line can miss a circle, cut it at two points, or just touch it at one point. The line that just touches is the tangent, and the point where it touches is the point of contact.

Definition

Three facts answer most one-mark questions:

A tangent meets the circle at exactly one point. A line meeting it at two points is a secant.
The number of tangents depends on where the point is: none from inside, one from a point on the circle, two from outside.
The tangents at the two ends of a diameter are parallel, and the gap between them is the diameter, $2r$ .

Situation	Fact
A line touching the circle at one point	A tangent
A line cutting the circle at two points	A secant
Tangents from a point inside the circle	None
Tangents from a point on the circle	Exactly one
Tangents from a point outside the circle	Exactly two
Tangents at the two ends of a diameter	Parallel to each other
Distance between two parallel tangents	The diameter, 2r It is the diameter, not the radius: a circle of radius 4 cm has parallel tangents 8 cm apart.

The tangent at a point is the one line through it that is perpendicular to the radius there.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Circles · Tangent and Radius

The tangents drawn at the extremities of the diameter of a circle are always:

Giving the radius as the gap between parallel tangents

Parallel tangents touch the circle at opposite ends of a diameter, so the gap is

2r

. For a radius of 3 cm the answer is 6 cm, not 3 cm.

Two tangents from a point on the circle

Only a point outside the circle has two tangents. A point on the circle has exactly one, and a point inside has none.

Drill 4 more on what a tangent is and how many there are

Concept 2 of 3: Length of a tangent from a right triangle

Join the centre to the point of contact and to the outside point. The radius meets the tangent at 90 degrees, so the three lines make a right triangle, and Pythagoras gives whichever side is missing.

Definition

The radius to the point of contact is perpendicular to the tangent.
The right triangle has the radius $r$ , the tangent length $t$ , and the distance $d$ from the centre to the outside point.
$d$ is the hypotenuse, the longest side, because it is opposite the right angle.
So $t^2 = d^2 - r^2$ , and $d^2 = r^2 + t^2$ .

Tangent length

t = \sqrt{d^2 - r^2} \qquad d = \sqrt{r^2 + t^2}

tlength of the tangent from the outside point
ddistance from the outside point to the centre
rradius of the circle

Worked example

A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Circles · Tangent and Radius

A tangent is drawn from a point at a distance of 17 cm of circle C(0, r) of radius 8 cm. The length of its tangent is

Adding the squares when one is the hypotenuse

The distance to the centre is the longest side. With distance 13 cm and radius 5 cm the tangent is

\sqrt{169 - 25} = 12

cm; adding gives

\sqrt{194}

, which is wrong.

Subtracting the lengths instead of their squares

With distance 13 cm and radius 5 cm,

13 - 5 = 8

cm is not the tangent. Square, subtract, then take the square root.

Drill 5 more on length of a tangent from a right triangle

Concept 3 of 3: Angles in the radius and tangent triangle

The right angle sits at the point of contact, so the other two angles of the triangle share the remaining 90 degrees. When one angle is 30 or 60 degrees, a simple ratio gives every side.

Definition

The angle between the radius and the tangent is $90^\circ$ , so the angle at the centre and the angle at the outside point add to $90^\circ$ .
In a right triangle: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ , $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$ .
$\sin 30^\circ = \cos 60^\circ = \frac{1}{2}$ and $\tan 45^\circ = 1$ . So in a 30, 60, 90 triangle the side opposite $30^\circ$ is half the hypotenuse.
Angles on a straight line add to $180^\circ$ : if QOR is a diameter, $\angle POQ = 180^\circ - \angle POR$ .

Angles at the point of contact

\angle OQP = 90^\circ \qquad \angle QOP + \angle OPQ = 90^\circ \qquad \sin 30^\circ = \cos 60^\circ = \tfrac{1}{2}

Worked example

PA is a tangent at A to a circle with centre O and radius 4 cm. If

\angle AOP = 60^\circ

, find

\angle APO

and OP.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Circles · Tangent and Radius

PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such

\angle POR = 120^\circ

, then

\angle OPQ

Putting the right angle at the wrong corner

The

90^\circ

angle is at the point of contact, between the radius and the tangent. It is never at the centre or at the outside point.

Using sin where cos is needed

If the angle at the centre is

60^\circ

, the radius is the side next to it, so use

\cos 60^\circ = \frac{\text{radius}}{\text{hypotenuse}}

. The radius is opposite the angle at the outside point.

Drill 2 more on angles in the radius and tangent triangle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Length of a tangent from a right triangle
Tangent length
$t = \sqrt{d^2 - r^2} \qquad d = \sqrt{r^2 + t^2}$
Angles in the radius and tangent triangle
Angles at the point of contact
$\angle OQP = 90^\circ \qquad \angle QOP + \angle OPQ = 90^\circ \qquad \sin 30^\circ = \cos 60^\circ = \tfrac{1}{2}$

Reference tables (1)

What a tangent is and how many there are7 rows

Situation	Fact
A line touching the circle at one point	A tangent
A line cutting the circle at two points	A secant
Tangents from a point inside the circle	None
Tangents from a point on the circle	Exactly one
Tangents from a point outside the circle	Exactly two
Tangents at the two ends of a diameter	Parallel to each other
Distance between two parallel tangents	The diameter, 2r It is the diameter, not the radius: a circle of radius 4 cm has parallel tangents 8 cm apart.

The tangent at a point is the one line through it that is perpendicular to the radius there.

Watch out for (6)

Giving the radius as the gap between parallel tangents
→ What a tangent is and how many there are
Two tangents from a point on the circle
→ What a tangent is and how many there are
Adding the squares when one is the hypotenuse
→ Length of a tangent from a right triangle
Subtracting the lengths instead of their squares
→ Length of a tangent from a right triangle
Putting the right angle at the wrong corner
→ Angles in the radius and tangent triangle
Using sin where cos is needed
→ Angles in the radius and tangent triangle

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