Tangents to a Circle: Quick Revision

A brief note on Tangents to a Circle. Read it before a test. Try each question at the end before you open its answer.

The tangent and the radius

• A tangent meets a circle at exactly one point, the point of contact.
• A line at distance $d$ from the centre of a circle of radius $r$ is a secant if $d < r$, a tangent if $d = r$, and outside if $d > r$.
• The tangent at a point is perpendicular to the radius through that point.
• From a point inside a circle, no tangent. From a point on it, one. From a point outside, two.

Two tangents from an outside point

• The two tangents from one outside point T are equal: $TP = TQ$.
• In the right triangle OPT, $PT^2 = OT^2 - OP^2$.
• OT bisects the angle between the tangents, and it is the perpendicular bisector of the chord PQ.
• $\angle POQ + \angle PTQ = 180^\circ$.

Circles inside figures

• Triangle: the tangents from one corner are equal, and each side is the sum of two tangent lengths.
• Right triangle with the right angle at B: $r = \dfrac{AB + BC - AC}{2}$.
• Four-sided figure round a circle: $AB + CD = BC + DA$. A parallelogram round a circle is a rhombus.
• Two circles with one centre: a chord of the larger that touches the smaller has half-length $\sqrt{R^2 - r^2}$.

A worked example

A tangent from a point T is $24$ cm long. T is $25$ cm from the centre. Find the radius, and the angle $\angle POQ$ if the angle between the tangents is $74^\circ$.

1. The radius. $r^2 = 25^2 - 24^2 = 49$, so $r = 7$ cm.
2. The angle at the centre. $\angle POQ = 180^\circ - 74^\circ = 106^\circ$.

Check. In the right triangle OPT, $\angle PTO = 37^\circ$ and $\angle POT = 53^\circ$. Together with the right angle, $37^\circ + 53^\circ + 90^\circ = 180^\circ$.

Practice with new numbers

A program to try

The program checks three ideas on numbers: the length of a tangent, the rule for the sides of a four-sided figure round a circle, and the radius of the circle inside a right triangle.

Mistakes to avoid

• Adding the squares to find a tangent. Subtract $r^2$ from $OT^2$.
• Forgetting the right angle. Mark it every time a tangent is drawn.
• Saying all tangents are equal. Only the two from the same outside point are.
• Using $r = \dfrac{a + b - c}{2}$ for any triangle. It works for a right triangle only, with $c$ the hypotenuse.

Check yourself

1. A point is $17$ cm from the centre of a circle of radius $15$ cm. How long is the tangent from the point?

2. The angle between two tangents is $40^\circ$. What is the angle between the two radii to the points of contact?