Parallel lines and proportional parts
Answers are for parents and teachers.
1
DE ∥ BC with D on AB and E on AC: AD = 6, DB = 3, AE = 8. Find EC, AC and the ratio DE : BC.
6/3 = 8/EC, so EC = 4 and AC = 12. DE : BC = AD : AB = 6 : 9 = 2 : 3.
2
D is on AB and E on AC: AD = 5, DB = 7.5, AE = 4, EC = 6. Is DE ∥ BC? If DE = 8, find BC.
5/7.5 = 2/3 and 4/6 = 2/3: yes. AB = 12.5, DE/BC = AD/AB = 5/12.5 = 2/5, so BC = 8 × 5/2 = 20.
3
Three parallel lines cut two transversals: AB = 4, BC = 6 and PQ = 10. Find QR and PR.
4/6 = 10/QR gives QR = 15, so PR = 25.
4
A sloping roof beam rests on three vertical posts 2 m and 3 m apart on the ground. The beam between the first two posts is 2.6 m. Find the beam between the last two posts.
2/3 = 2.6/x, so x = 3.9 m.
5
In quadrilateral ABCD, X is on AB, Y on AC and Z on AD with XY ∥ BC and YZ ∥ CD. AX = 3, XB = 5 and AZ = 9. Find ZD and explain why XZ ∥ BD.
AX/XB = AY/YC = AZ/ZD, so 3/5 = 9/ZD and ZD = 15. Since AX/XB = AZ/ZD, the converse gives XZ ∥ BD.