At a glance. One lesson of about 60 minutes, with no calculator. It assumes Pythagoras and congruence. Students prove that the perpendicular bisects a chord, that a tangent meets the radius at a right angle and that tangents from a point are equal, then use them with Pythagoras and angle sums.
Calculator: no calculator. No calculator is needed: every length here is a whole number or a surd, and the point is the reasoning.
What students will learn
Know
- The perpendicular from the centre to a chord bisects it.
- A tangent is perpendicular to the radius at the point of contact.
- Two tangents from a point have equal length.
Understand
- Each result is a consequence of equal radii and right-angled triangles, not a rule to memorize.
- A tangent is the limiting case of a chord moving away from the centre.
Do
- Write a proof with a reason for each step.
- Use Pythagoras for chord, tangent and distance problems.
- Use angle sums in the quadrilateral formed by two tangents and two radii.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: chord and tangent properties and Pythagoras (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe how the chord length changes with the distance and why the tangents are equal.
- Criterion C. Communicating: a proof with each line justified in words (worked example 1), and labelled diagrams with right-angle marks.
- Criterion D. Applying mathematics in real-life contexts: the window bar (the hook and worked example 2) and the path across a circular pond (extra questions).
The one thing to do with the figure
Slide until the chord vanishes. In the first figure, set the radius, then move the distance up one step at a time. Ask: at which step does the chord become a single point? What is that line called, and what do you notice about it and the radius?
In the second figure, ask students to predict which of PT and PT' is longer before they move anything.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the bar across the window. Collect votes. |
| 10 min | First figure; settle the hook with worked example 2. |
| 12 min | Key card: the three proofs; worked example 1 done together; the warning card. |
| 5 min | Second figure; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to what M is. |
| 13 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 40; 360°; the angles are equal. |
|---|---|
| Hook | 30 cm, longer than the 17 cm radius. |
| Worked examples | 90°; 30 cm; 21 cm. |
| Faded example | 56 cm. |
| 1. Why the radius meets a chord at a right angle | B. A calls OM a radius; C quotes an angle sum that fixes nothing; D uses the tangent result for a chord. |
| 2. Chord length from its distance | 70 cm. |
| 3. Length of a tangent | 12 cm. |
| 4. Distance from the centre | 5 cm. |
| 5. Area of a kite from two tangents | 48 cm². |
| 6. Angle between two tangents | 130°. |
| 7. A chord tangent to an inner circle | 24 cm. |
| 8. A chord with algebra | 3 cm. |
| 9. Equal tangents with algebra | C. A reports PA; B and D are sign slips. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Called OM a radius, quoted an angle sum, or used the tangent result for a chord. | Ask: is M on the circle? Then what is OM? |
| 35 or 24 in question 2 | Gave the half chord, or doubled the distance. | Mark half the chord on the sketch, then say what the whole chord is. |
| 24 or 6 in question 3 | Added or subtracted lengths. | Mark the right angle at T and the hypotenuse OP before any number. |
| 1 or 17.7 in question 4 | Subtracted lengths, or added squares. | Ask: which of the three lengths is the longest side of the triangle? |
| 24 or 60 in question 5 | Gave one triangle, or multiplied radius by OP. | Find PA first, then ask how many triangles OAPB is made of. |
| 40 or 310 in question 6 | Used one triangle's other acute angle, or forgot the right angles. | List the four angles of the quadrilateral: 90, 90, 50 and ? |
| 12 or 16 in question 7 | Gave the half chord, or doubled the difference. | Ask what the inner radius is, in terms of the chord. |
| 4 or 8 in question 8 | Answered with the half chord or the whole chord. | Underline what the question asks for before solving. |
| A, B or D in question 9 | Gave a length, or made a sign slip. | Write the equation PA = PB first, then solve line by line. |
Questions worth asking
- Why does the proof of result 1 not need a calculator or a measurement? Listen for “it uses only equal radii and Pythagoras”.
- What is a tangent, as a limiting case of a chord? A chord whose two ends have come together.
- Why must the hypotenuse of triangle OTP be OP? It is opposite the right angle at T.
- How can you use result 3 to find an unknown in an equation? Set the two tangent expressions equal.
Differentiation
A way in for anyone stuck
- Give the proof of result 1 with the reasons on separate cut-out strips to put in order.
- Provide a diagram with the right angle already marked and the hypotenuse highlighted.
- Restrict to whole-number triples: 3, 4, 5; 5, 12, 13; 8, 15, 17.
Stretch
- Prove that the perpendicular bisector of any chord passes through the centre of the circle.
- Two tangents from P to a circle of radius r make an angle of 60° at P. Find OP in terms of r. (OP = 2r.)
Students learning in English as an additional language
“Chord” is a straight line joining two points on a circle. “Tangent” touches the circle at one point and does not cross it. “Perpendicular” means at a right angle. “Bisect” means cut exactly in half. Draw each on the board and keep the pictures up.
| Word | Plain meaning | Example |
|---|---|---|
| chord | a straight line joining two points on a circle | the bar across a window |
| tangent | a line that touches a circle at exactly one point | a ruler laid against a wheel |
| radius | a line from the centre to the circle, or its length | a spoke of a wheel |
| bisect | cut exactly in half | the line from the centre to the middle of a chord |
| perpendicular | at a right angle to | the radius at the tangent |
| point of contact | the one point where a tangent touches the circle | T in the figure |
Sentence starters for explaining aloud or in writing:
- OA = OB because …
- The triangles are congruent because …
- The angle is 90° because the tangent …
Two things not to say
“That is just a circle fact.” Ask: how do we know? What is the reason?
“Draw it and measure.” A measurement is not a proof.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ChordA circle has radius 10 cm and a chord of length 12 cm. How far is the chord from the centre?
8 cm. - Earlier work (Year 9)A right-angled triangle has sides 9 and 40. Find the hypotenuse.
41.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA circular pond has radius 20 m. A straight path crosses it, with its nearest point 12 m from the centre. How long is the path across the pond?
32 m. - Working backA tangent from P to a circle has length 20 cm and the radius is 21 cm. Find OP.
29 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why the line joining the centre to the point P outside a circle bisects the angle between the two tangents.
Triangles OTP and OT'P are congruent (RHS), so the angles at P are equal. - OpenFind a circle radius, a chord length and a distance that are all whole numbers, other than 13, 24, 5.
For example radius 25, chord 48, distance 7.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? A compass and a ruler give chords and tangents to measure. No calculator is used.