Intersecting-chord/secant/tangent length relationships | Year 11

01 / LEARN

The idea, step by step

Inside the circle

When chords AB and CD intersect at E inside a circle, $A E \times E B = C E \times E D$ . Each product uses the two parts of one complete chord, measured from the intersection. The theorem can be proved using pairs of similar triangles with a vertical angle and a pair of equal inscribed angles.

Write the two products using point names before substituting numbers. This prevents accidentally multiplying one part from each unrelated chord. All the distances are ordinary positive lengths.

Outside the circle

For secants PAB and PCD from an external point P, where A and C are the near circle intersections, $P A \times P B = P C \times P D$ . PB and PD are whole distances from P to the far intersections, not just the segments inside the circle.

If PT is tangent at T, then $P T^{2} = P A \times P B$ . As an alternative check, the same common value is $P O^{2} - r^{2}$ for centre O and radius r. Set up the geometry first; a negative squared tangent length signals impossible or incorrectly interpreted data.

A common trap

Multiplying the outside and inside parts of a secant.

Instead: The external-secant theorem uses outside times whole, including the outside part twice in the two factors.

02 / WORKED EXAMPLES

See the thinking

Example 1

Intersecting chords

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

The internal-chord equation is $3 \times 12 = 4 \times E D$ .
Divide 36 by 4.

= 9

cm.

Example 2

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.

The whole secant from P to the far intersection is $4 + 5 = 9$ cm.
$P T^{2} = 4 \times 9 = 36$ .
Take the positive square root.

= 6

cm.

Check your understanding

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

Show answer

6 units.

$P T^{2} = 3 \times 12 = 36$ .

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF 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 ↗

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