Circles · Class 10 Mathematics: quick revision, 35 bites

Miss, cut or touch

Only three possibilities

A line and a circle share no point, two points, or exactly one.

Takeaway. There is no fourth case.

Point of contact

The single meeting place

The one point a tangent shares with its circle.

Takeaway. Every theorem in this chapter lives at that point.

The tangent as a limit

A secant that ran out of room

Slide a secant outwards and its two crossing points move together.

Takeaway. A tangent is a secant with its points merged.

Tangent is perpendicular to radius

The one theorem

Takeaway. Almost everything else follows from this.

By contradiction, in three lines

How the proof runs

Takeaway. No right angle is assumed anywhere — it is the conclusion.

Mark it at the contact point

Where the square goes

The right angle is at P, where the tangent touches — not at the centre and not at the outside point.

Takeaway. Marking it wrongly loses the whole question.

The tangent length

Pythagoras arrives

Takeaway. The right angle is what makes this legal.

OP is the hypotenuse

Which side is longest?

The right angle sits at the point of contact, so the side FACING it — from the centre to the outside point — is the hypotenuse.

Takeaway. Centre to outside point, always.

r = 5, d = 13

A first calculation

Takeaway. The 5-12-13 triple, chosen on purpose.

r = 7, tangent 24

Working backwards

Takeaway. Now the unknown is the hypotenuse, so the terms are added.

How many tangents

In none, on one, out two

Takeaway. Where the point is decides everything.

Every line must leave

Why none from inside

A line through an inside point has to cross the circle on both sides, so it is a secant.

Takeaway. Touching once is impossible from within.

Tangents at a diameter's ends

A free result

Both are perpendicular to the same straight line, the diameter.

Takeaway. So those two tangents never meet.

The converse

Running the theorem backwards

If a line through a point of the circle is perpendicular to the radius there, it IS a tangent.

Takeaway. This is how you PROVE something is a tangent.

Shortest distance

The closest point

$OP$ is the perpendicular distance from the centre to the tangent, so no point of that line is nearer to $O$ than the point of contact.

Takeaway. The contact point is the closest point of the line.

What 'tangent length' means

Length, not line

The tangent LINE is infinite.

Takeaway. A finite segment, taken from a named point.

Quoting the theorem

Say the words

Write 'the tangent at a point of a circle stands at right angles to the radius drawn to that point' before using the right angle.

Takeaway. The statement itself carries marks.

Where the word comes from

Latin for touching

'Tangent' comes from tangere, to touch — the same root as 'tangible'.

Takeaway. The name says exactly what it does.

Theorem 2

Exactly two, and equal

From a point outside a circle, exactly two tangents can be drawn.

Takeaway. The equality is what makes tangent figures tractable.

The three ingredients

Why they must be equal

Takeaway. Right angle, Hypotenuse, Side — RHS.

Naming the criterion

Not SSS, not SAS

Only two sides are known equal at the start, and the equal angle is not between them.

Takeaway. Say RHS explicitly.

What CPCT gives next

Three free results

$\angle OPA = \angle OPB$: $OP$ bisects the angle between the tangents.

Takeaway. One congruence, several conclusions.

OAPB

A kite in disguise

$OA = OB$ and $PA = PB$, so $OAPB$ is a kite with $OP$ as its axis of symmetry.

Takeaway. The symmetry explains every result at once.

The length

One calculation, both tangents

Takeaway. Theorem 2 means you never compute it twice.

r = 7, OP = 25

A worked length

Takeaway. And PB is 24 as well, without further work.

The quadrilateral result

Wrapped around a circle

Takeaway. Opposite sides, equal sums.

Four applications of Theorem 2

Where that comes from

Each vertex is an external point, so the two tangent lengths from it are equal.

Takeaway. Nothing new is needed — only Theorem 2, four times.

Which sides to pair

Opposite, not adjacent

It is $AB + CD = BC + DA$, pairing sides that face each other.

Takeaway. Draw the quadrilateral and look across it.

Circumscribing parallelograms

Forced into a rhombus

A parallelogram already has $AB = CD$ and $BC = DA$.

Takeaway. So a non-square rectangle can never circumscribe a circle.

s minus the opposite side

Where the incircle touches

Takeaway. s is the semiperimeter, half the perimeter.

Sides 14, 10, 12

A triangle worked out

$s = 18$, so the tangent lengths are $4$, $8$ and $6$.

Takeaway. The three splits must rebuild the sides.

At the point and at the centre

Supplementary angles

Takeaway. Two right angles in the kite use up the rest.

The method

Label first, calculate later

Takeaway. Most questions dissolve at the labelling step.

Where the marks are

Quote the theorems

State Theorem 1 for the right angles and Theorem 2 for the equal tangents, by name, before using them.

Takeaway. The statements carry marks of their own.

Equal tangents outside geometry

A ball in a corner

A ball resting against two walls touches each at the same distance from the corner — the same theorem, in three dimensions.

Takeaway. Symmetry does the work.