Practice Intercept theorems
The intercept theorems relate corresponding lengths when parallel lines cut two rays from a common point.
They help you calculate inaccessible distances through proportions.
Exercises with solutions
The intercept theorems apply when two rays are cut by parallel lines. Then the sections on one ray relate to the corresponding sections on the other: SA : SB = SA' : SB'. The sections between the parallels are in the same ratio.
Exercise 1
Two rays are cut by parallel lines. SA = 3 cm, SB = 5 cm and SA' = 6 cm. How long is SB'?
Show solution and working
10 cm
SA : SB = SA' : SB', so 3 : 5 = 6 : SB'. Therefore SB' = 6 cm · 5 ÷ 3 = 10 cm.
Exercise 2
SA = 4 cm, SA' = 10 cm and the parallel at A is 6 cm long. How long is the parallel at A'?
Show solution and working
15 cm
The ratio is 10 : 4 = 2.5. So 6 cm · 2.5 = 15 cm.
Exercise 3
How do you tell that the intercept theorems may be used?
Show solution and working
from two parallel lines cutting two rays
Without parallel lines the intercept theorems do not apply.
Exercise 4
SA = 2 cm, SB = 3 cm and SB' = 7.5 cm. How long is SA'?
Show solution and working
5 cm
2 : 3 = SA' : 7.5, so SA' = 2 cm · 7.5 ÷ 3 = 5 cm.
Common mistake
Pairing the sections incorrectly. Always put sections on the same ray into a ratio, not a section of one ray with one of the other.
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