Chords, secants and tangents of a circle
Answers are for parents and teachers.
1
Chords AB and CD cross at E inside a circle: EA = 6, EB = 5, EC = 3. Find ED and CD.
6 × 5 = 30 = 3 × ED, so ED = 10 and CD = 3 + 10 = 13.
2
Secants from E outside: EA = x, AB = 4, EC = 5, ED = 12. Find x and EB.
x(x + 4) = 60, x² + 4x − 60 = 0, (x − 6)(x + 10) = 0, so x = 6 and EB = 10.
3
The tangent from E has length 15 and the outer part of a secant is 9. Find the whole secant and the chord DC.
EC = 225/9 = 25 and DC = 25 − 9 = 16.
4
D is on AB and E is on AC: AD = 2, AB = 21, AE = 6, AC = 7. Prove that D, B, C, E lie on one circle.
AD × AB = 2 × 21 = 42 and AE × AC = 6 × 7 = 42. The products are equal, so by the converse the four points are concyclic.
5
A circle has radius 6 and E is 10 from the centre. Find the tangent length, and check it with the secant through the centre.
Tangent² = 100 − 36 = 64, so 8. The secant through the centre has ED = 10 − 6 = 4 and EC = 10 + 6 = 16, and 4 × 16 = 64 = 8² ✓.