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Teach Circles in five steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

  • A straight fence can touch a round flowerbed without cutting across it.
  • A wheel’s bottom radius meets level ground at a right angle.
  • A round pipe touching both sides of a V-shaped support sits along its angle bisector.

What to teach

Recognise tangents

Tangent to a CircleA straight line that touches a circle without crossing it.Theorem 10.1: Tangent and RadiusThe contact radius meets the tangent squarely.

Compare and calculate

Theorem 10.2: Tangents from an External PointTwo tangents starting together have equal lengths up to contact.Length of a TangentUse the contact right triangle to find lengths.Quadrilateral Circumscribing a CircleOpposite sides have equal totals.

How to teach it

Recognise tangents

Lines crossing, touching and missing a circle; two tangents meet at an outside point.

1. Tangent to a Circle

A straight line that touches a circle without crossing it.

Show
Look at the left diagram and count each line’s shared points with the circle. On the right, follow the lines from P to Q and R.
Ask
Looking at these lines, how many shared points do you predict for each?
Say
The tangent shares one point with the circle; the secant shares two. The outside point P has two tangents, touching separately at Q and R.
Used in
A tangent fence skirts a circular flowerbed while keeping its planting space intact.
If your child says…“A tangent crosses the circle twice.”
Ask
How many shared points can you count on each line?
What’s true
Crossing twice makes a secant; a tangent only touches.
Radius OT meets a tangent at T; OA extends beyond the circle to A.

2. Theorem 10.1: Tangent and Radius

The contact radius meets the tangent squarely.

Show
Find OT in the diagram. Follow OA through B to the tangent.
Ask
Why must OT meet the tangent at a right angle?
Say
Every other tangent point is farther from O. So OT is shortest, and the shortest distance meets a line perpendicularly.
Used in
A circular wheel’s axle sits one radius above level ground, helping determine its height.
If your child says…“The right angle belongs at the centre.”
Ask
Which end of OT reaches the tangent?
What’s true
The right angle sits where the radius reaches the touching line.

Compare and calculate

Two tangents form right triangles sharing OP, with equal radii marked.

3. Theorem 10.2: Tangents from an External Point

Two tangents starting together have equal lengths up to contact.

Show
Find both right triangles in the diagram. Trace their shared hypotenuse OP and equal radii.
Ask
Why must PQ and PR have equal lengths?
Say
The triangles match by RHS, so their tangent sides are equal.
Used in
A round pipe’s centre follows a V-shaped support’s angle bisector, helping position it centrally.
If your child says…“Each triangle gets the whole outside angle.”
Ask
Where are the two equal angles at P within the whole angle?
What’s true
Each matching triangle contains half the angle between the tangents at P.

x2+32=52x^2 + 3^2 = 5^2

x2=25−9x^2 = 25 - 9

x2=16x^2 = 16

4. Length of a Tangent

Use the contact right triangle to find lengths.

Show
Write these lines. Say, “The radius is 3 cm; the centre is 5 cm away. Call the tangent length x.”
Ask
What positive length has square 16?
Say
The tangent measures 4 cm because 4 squared is 16.
Used in
Tangent lengths help size straight belt runs from pulleys to guides.
If your child says…“The tangent must be the longest side.”
Ask
Which side faces that right angle?
What’s true
The centre-to-outside-point side faces the contact right angle, so it is longest.
Quadrilateral surrounding a circle, with equal tangent pieces labelled at each corner.

5. Quadrilateral Circumscribing a Circle

Opposite sides have equal totals.

Show
Look at the diagram of quadrilateral ABCD around the circle. Find the two equal tangent pieces from each corner.
Ask
From the equal tangent pieces, which side totals do you predict will match?
Say
Each opposite-side total contains one a, one b, one c and one d. The two sums of opposite sides are equal: AB+CD=AD+BCAB + CD = AD + BC.
Used in
These totals determine missing frame lengths around circular mirrors.
If your child says…“Neighbouring sides must have equal totals.”
Ask
Which pairing uses a, b, c and d exactly once?
What’s true
Each opposite-side total contains one tangent piece from every corner.

A real problem

A mountain helped measure Earth’s radius.

Al-Biruni used a mountain’s height to determine Earth’s radius.

He carried out the measurement at Nandana.

His method treated the sightline to the horizon as a tangent to Earth.

He completed the Tahdid, which records this method, in 1025.

University of St AndrewsEncyclopaedia IranicaEncyclopaedia Iranica

Clear distant horizon beyond a rocky hilltop.

Concept Lab · Common misconception

The tempting mistake in Circles

Open standalone misconception page →

Does a line through one point on a circle have to be a tangent?

Wrong

Any line drawn through a marked point on the circle is a tangent.

Why it feels right. I count the one marked contact point and overlook a second intersection.

Right

A tangent touches the circle at exactly one point. A secant crosses it at two.

Worked example

The horizontal secant meets the circle twice. The vertical tangent meets it once.
  1. At the rightmost point, draw a horizontal and a vertical line.
  2. The horizontal line crosses the circle again on the left.
  3. The vertical line has one contact and is perpendicular to the radius there.

Ask your child five quick questions.

One per step. The answer and the reason are given.

  1. What is a line crossing a circle at two separate points called?

    A secant. Two shared points make a secant; touching once makes a tangent.

  2. Which corner of triangle OTA is a right angle?

    At T, the contact point. The contact radius meets the touching line squarely.

  3. The tangents meet at 60°. What angle does either triangle have at P?

    30°. The line joining P to the centre divides the outside angle equally.

  4. Which distance is longest?

    Centre to outside point. It faces the contact right angle.

  5. Which side is opposite AB in quadrilateral ABCD?

    CD. Opposite sides include every corner’s contribution.

15 more questions with answers, 5 with pictures