Parallel lines and equal parts: transversals cut in equal segments
Mathematics Prep 2 Term 1, Unit 3, Lessons 3–4
GoalUse equal parts cut by parallel lines on one transversal to find equal parts on another, and use the midpoint idea in a triangle.
New words
transversal · a line that cuts across several other linesparallel lines · lines in one plane that never meet; equally spaced lines are an easy picturemidpoint · the point that cuts a segment into two equal partsequal parts · segments of the same length, marked with the same number of ticks
Explanation
Think of ruled notebook paper: the lines are parallel and equally spaced. Draw any slanted line across several lines and the parts between the lines are equal. This is the first rule: if parallel lines cut equal parts on one transversal, they cut equal parts on every transversal. In the picture AB = BC on the left, and the three parallel lines AD, BE and CF give DE = EF on the right. The lengths on the two transversals are different, but the parts on each are equal among themselves. The rule needs the lines to be parallel; lines that fan out do not work. A second rule follows in a triangle: if D is the midpoint of AB and a line through D is parallel to BC, then it meets AC at its midpoint E. Idea: draw a line through A parallel to BC. The three parallels cut AB into equal parts AD and DB, so they cut AC into equal parts AE and EC. Both conditions are needed: D must be the midpoint, and DE must be parallel to BC.
Worked examples
Parallel lines AD, BE and CF cut the transversal AC with AB = BC = 3 cm. On the other transversal DE = 2x + 1 and EF = 9. Equal parts on one give equal parts on the other, so 2x + 1 = 9, 2x = 8 and x = 4. Then DE = 9 cm and DF = 18 cm.
Three equally spaced parallel lines cut one path into two equal parts. On a second path the two parts are 5x − 4 and 3x + 6 metres. Since they are equal, 5x − 4 = 3x + 6, so 2x = 10 and x = 5. Each part is 5 × 5 − 4 = 21 m, and the whole second path is 42 m. Check the other part: 3 × 5 + 6 = 21 m.
In Δ ABC, D is the midpoint of AB and a line through D parallel to BC meets AC at E. Then E is the midpoint of AC. If AE = 3x − 2 and EC = x + 6, then 3x − 2 = x + 6, so x = 4, AE = 10 cm and AC = 20 cm.
Class activity
Ruled paper test. Draw two different slanted lines across a page of ruled paper (or draw five equally spaced parallel lines). Measure the parts on each slanted line with a ruler. Are the parts on each line equal? Now draw lines that fan out from one point and measure again: what changes?
Practice
1
Parallel lines cut AB = BC on one transversal. On another transversal DE = 4x − 3 and EF = 2x + 5. Find x and DE.
4x − 3 = 2x + 5, so 2x = 8 and x = 4. DE = 4 × 4 − 3 = 13.
2
Four parallel lines cut one transversal into three equal parts of 5 cm each. The whole second transversal is 27 cm. How long is each part of it?
The parts on the second are equal too: 27 ÷ 3 = 9 cm each.
3
In Δ ABC, M is the midpoint of AB and MN ∥ BC with N on AC. If AN = 3x − 1 and NC = x + 7, find x and AC.
N is the midpoint of AC, so 3x − 1 = x + 7, x = 4. AN = 11 and AC = 22.
4
Three lines that are not parallel cut AB = BC on one transversal. Can we say they cut equal parts on every transversal?
No. The rule needs parallel lines. Lines that fan out from a point cut longer parts further away.
5
Mariam: ‘D is on AB with AD = 3 and DB = 6, DE ∥ BC, so E is the midpoint of AC.’ What is wrong?
D is not the midpoint of AB (3 ≠ 6), so the rule does not apply and E is not the midpoint.
