Intersecting Chords, Secants & Tangents | Year 11

01 / LEARN

Understand the idea.

Inside the circle

For two chords intersecting at E inside a circle, multiply the two parts of each complete chord: $A E \times E B = C E \times E D$ . Write the point-name equation first so that each product follows one straight chord through E.

The relationship follows from similar triangles: a vertical-angle pair and angles subtending the same chord supply equal angles. Use ordinary positive segment lengths, and keep the two parts of one chord together rather than multiplying unrelated adjacent segments.

Outside the circle

For secants from an external point P, multiply the outside distance by the whole distance to the far intersection: $P A \times P B = P C \times P D$ . When an inside segment is given, add it to the outside segment before forming the product.

A tangent from the same point satisfies $P T^{2} = P A \times P B$ . The common value also equals $P O^{2} - r^{2}$ , with centre O and radius $r$ . A negative squared length cannot describe a real tangent and signals impossible data or a mistaken segment interpretation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Intersecting chords

Chords AB and CD meet at E. AE $= 3$ cm, EB $= 12$ cm and CE $= 4$ cm. Find ED.

Worked solution

1
Write the intersecting-chord relation.
$A E \times E B = C E \times E D$
Both chords meet at the same internal point E.
2
Substitute the three known parts.
$3 \times 12 = 4 \times E D$
Each product uses the two parts of a single chord.
3
Solve for the missing segment.
$E D = \frac{36}{4} = 9 cm$
Dividing by the known positive length isolates ED.

Answer

ED $= 9$ cm.

EXAMPLE 02

Use the whole secant

A secant from P has outside part 4 cm and inside part 5 cm. A tangent PT is drawn from P. Find PT.

Worked solution

1
Calculate the whole secant distance.
$whole length = 4 + 5 = 9 cm$
The whole distance begins at P and includes both the outside and inside parts.
2
Apply the tangent–secant relation.
$P T^{2} = 4 \times 9 = 36 {cm}^{2}$
The squared tangent equals outside distance times whole secant distance.
3
Take the positive square root.
$P T = \sqrt{36} = 6 cm$
A tangent segment has a positive length.

Answer

PT $= 6$ cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

A secant has outside length 3 and whole length 12. Find the tangent length from the same point.

Show answer & reasoning

6 units.

The tangent–secant relation gives $P T^{2} = 3 \times 12 = 36$ . Taking the positive square root gives a tangent length of $6$ units.

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Chords meet at $E$ with segment lengths 2 and 15 on one chord, and 5 and $x$ on the other. Find $x$ .
2 marks
A tangent has length 10. A secant from the same point has outside length 4. Find its whole length.
2 marks
A secant has outside length 2 and whole length 18. Another has outside length 3. Find the second secant's inside length.
3 marks
A tangent is 12 cm long. The external point is 13 cm from the centre. Find the circle radius.
2 marks
A student uses $P T^{2} = 3 \times 8$ when a secant has outside part 3 and inside part 8. Correct the calculation.
3 marks
Intersecting chords have parts $x$ and $x + 3$ on one chord, and 2 and 14 on the other. Find $x$ .
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Chords meet at $E$ with segment lengths 2 and 15 on one chord, and 5 and $x$ on the other. Find $x$ .
2 marks
$2 \times 15 = 5 x$ , hence $x = 6$ .
$x = 6$ .
A tangent has length 10. A secant from the same point has outside length 4. Find its whole length.
2 marks
$10^{2} = 4 L$ , so $L = 25$ .
25 units.
A secant has outside length 2 and whole length 18. Another has outside length 3. Find the second secant's inside length.
3 marks
$2 \times 18 = 3 L$ , so the second whole length is 12.
Its inside length is $12 - 3 = 9$ .
9 units.
A tangent is 12 cm long. The external point is 13 cm from the centre. Find the circle radius.
2 marks
$12^{2} = 13^{2} - r^{2}$ , so $r^{2} = 25$ .
5 cm.
A student uses $P T^{2} = 3 \times 8$ when a secant has outside part 3 and inside part 8. Correct the calculation.
3 marks
Use the whole secant length 11, not the inside length 8.
$P T^{2} = 3 (3 + 8) = 33$ , so $P T = \sqrt{33}$ .
Intersecting chords have parts $x$ and $x + 3$ on one chord, and 2 and 14 on the other. Find $x$ .
4 marks
$x (x + 3) = 28$ gives $(x + 7) (x - 4) = 0$ .
Reject $x = - 7$ because a length is positive.
$x = 4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

Intersecting-chord/secant/tangent length relationships

In this lesson

Understand the idea.

Inside the circle

Outside the circle

See the method in action.

Intersecting chords

Write the intersecting-chord relation.

Substitute the three known parts.

Solve for the missing segment.

Use the whole secant

Calculate the whole secant distance.

Apply the tangent–secant relation.

Take the positive square root.

Try it yourself.

Check your understanding

Keep practising