At a glance. One lesson of about 60 minutes. It assumes the circle angle theorems and similar triangles. Students prove and use AX × XB = CX × XD and PT² = PA × PB, and prove that a tangent is perpendicular to the radius by contradiction.
Calculator: no calculator. No calculator is used: every length in the questions is a whole number or has a whole-number square root.
What students will learn
Know
- AX × XB = CX × XD for two chords crossing at X.
- PT² = PA × PB for a tangent and a secant.
- A tangent is perpendicular to the radius at the point of contact.
Understand
- A product that does not change with direction is an invariant, and the similar triangles show why.
- The tangent result is the limit of the secant result as A and B come together.
Do
- Use the products to find a length, including by solving a quadratic.
- Prove the product results from similar triangles.
- Prove the tangent property by contradiction.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the product results and angle theorems (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students describe what changes and what stays the same as the chord or secant turns.
- Criterion C. Communicating: proofs with every reason written, including the angle pairs that make the triangles similar (key card, worked examples).
- Criterion D. Applying mathematics in real-life contexts: the two paths across the plaza.
The one thing to do with the figure
Make students predict, then reveal. Before touching the first figure, ask for the unknown piece, 9, and for the rule they used. Then show that a second guess rule (the sum) fails.
In the second figure, set the secant to 0° through the centre and have students check that PA × PB equals (OP − r)(OP + r).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the two paths. Collect predictions and reasons. |
| 12 min | First figure; key idea and the similar triangles; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, return to sums against products. |
| 18 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 | 20 cm; 15 cm; 33°. |
|---|---|
| Worked examples | 10; PB = 28, AB = 21, r ≈ 20.7; (OP − r)(OP + r) = OP² − r². |
| Faded example | 6. |
| Something to think about | A tangent: PT × PT = PT². |
| 1. What stays fixed for crossing chords | C. A and D match the sums; B keeps the ratio. |
| 2. A missing piece of a chord | 14 cm. 42 ÷ 3. |
| 3. A tangent and a secant | 10 cm. PB = 18. |
| 4. Crossing chords as an equation | 18 cm. x = 6. |
| 5. Radius from a tangent | 8 cm. √64. |
| 6. Angle chase with a tangent and a secant | 84°. 120 − 36. |
| 7. Distance between parallel chords | 7 cm. 12 − 5. |
| 8. Similar triangles from crossing chords | 15 cm. 6 × 10 ÷ 4. |
| 9. A tangent from a secant | B. A uses the chord; C uses Pythagoras; D gives PB. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Matched the sum of the pieces, or kept them in the same ratio. | Ask for the product of each pair and compare it with 6 × 8. |
| 2.6 or 21 in question 2 | Paired CX with XB, or multiplied AX by CX. | Write AX × XB on one side and CX × XD on the other before substituting. |
| 18 or 4 in question 3 | Gave PB, or subtracted the lengths. | Mark PB and AB on the sketch and ask which is the chord. |
| 6 or 12 in question 4 | Gave x, or one piece. | Underline what is asked: the length AB, the whole chord. |
| 22.7 or 2 in question 5 | Added the squares, or subtracted lengths. | Draw the right angle at T and label the hypotenuse OP. |
| 120 or 60 in question 6 | Gave the whole angle at T, or its supplement. | Mark the three angles at T: PTA, ATB and their sum PTB. |
| 17 or 14 in question 7 | Added the distances, or subtracted the chord lengths. | Draw both chords on the same side of O and find each distance. |
| 2.4 or 6.7 in question 8 | Inverted a ratio, or divided by the wrong side. | Match the vertices A to D, X to X, B to C before writing the ratio. |
| A, C or D in question 9 | Used the chord AB, used Pythagoras, or gave PB. | Say: the product uses the whole distance from P; then take the square root. |
Questions worth asking
- Why must AX × XB = CX × XD, not AX × CX? Listen for “the similar triangles pair AX with CX and XD with XB, so the cross-multiplication pairs the two pieces of one chord”.
- Why are triangles AXD and CXB similar? Listen for “same-arc angles and vertically opposite angles”.
- What is the tangent result a special case of? Listen for “the secant result when A and B coincide”.
- In the contradiction proof, what is the assumption and what is contradicted? Listen for “assume OT is not perpendicular; the tangent would meet the circle twice”.
Differentiation
A way in for anyone stuck
- Give a diagram with the four pieces labelled and two colours for the two chords.
- Provide a template: piece × piece = piece × piece, with boxes to fill.
- Restrict to questions 1 to 4 at first, with whole-number answers.
Stretch
- Prove that the lengths of the two tangents from a point are equal, in two ways: congruent triangles and PT² = OP² − r².
- Two circles intersect at A and B. A line through A meets the circles again at C and D. Show that angle CBD is the same for every such line.
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. “Secant” cuts the circle at two points. Draw all three in one circle and label them.
| Word | Plain meaning | Example |
|---|---|---|
| chord | a straight line joining two points on a circle | a path across a round plaza |
| tangent | a straight line that touches a circle at exactly one point | a ruler touching a coin at its edge |
| secant | a straight line that cuts a circle at two points | a path that crosses the plaza |
| similar triangles | triangles with the same angles and sides in the same ratio | a photograph and its enlargement |
| contradiction | a result that cannot be true at the same time as what you assumed | a line that is both a tangent and meets the circle twice |
Sentence starters for explaining aloud or in writing:
- The two pieces of one chord multiply to …
- The triangles are similar because …
- Suppose the tangent is not perpendicular to the radius. Then …
Two things not to say
“Just use the formula.” Ask which two triangles prove it.
“The chord is the whole distance from P.” Separate the chord from the secant's PB.
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.
- ChordsChords AB and CD cross at X with AX = 4, XB = 9 and CX = 6. Find XD.
6. - Earlier work (Year 9)Two similar triangles have sides in the ratio 2 : 7. A side of 8 cm in the small one is how long in the large?
28 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TangentFrom P the tangent length is 10 and the secant PAB has PA = 4. Find PB.
25. - RadiusOP = 13 and PT = 12 for a tangent PT. Find the radius.
5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why the product of the pieces of a chord through a fixed point does not depend on the chord.
By the theorem it equals r² minus d², which depends only on r and the distance d from the centre. - OpenA chord of a circle is cut by X into pieces 3 and 12. The circle has radius 10. How far is X from the centre?
√(100 − 36) = 8.
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? Two strings across a circle drawn in chalk give the pieces to measure.