When several parallel lines cut across two straight lines, the ratio of the segments on one line appears again, exactly, on the other.
The lengths themselves differ from line to line, but the parallel lines carry the ratio across unchanged.
Inside a triangle, a segment parallel to one side divides the other two sides in the same ratio, and the segment joining the midpoints of two sides is parallel to the third side and exactly half as long.
In the first picture the lengths a, b, c and d are all different, yet a : b and c : d always move together.
Parallel lines carry a ratio
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The lengths a, b, c, d of the segments cut from two lines by three parallels, and their ratios
2
👀 Parallel lines and segment ratios
①When three parallel lines ℓ₁, ℓ₂, ℓ₃ cut two lines, they cut segments a and b on one line and c and d on the other.
②Because the two lines lean differently, a and c, and b and d, have different lengths.
③Even so, a : b = c : d always holds. Changing the ratio in the picture changes all four lengths, but the two ratios move together.
A parallel line inside a triangle
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How a segment DE parallel to BC divides the other two sides of triangle ABC
0.5
Triangles and parallel lines
DE ∥ BC → AD : AB = AE : AC = DE : BC
A segment parallel to one side divides the other two sides in the same ratio, because △ADE and △ABC are similar.
Midpoint theorem
Midpoint theorem
D, E midpoints → DE ∥ BC, DE = 12BC
The segment joining the midpoints of two sides is parallel to the third side and half its length.
💡 How to read the midpoint theorem
①If D and E are the midpoints of AB and AC, then AD : AB = AE : AC = 1 : 2.
②So △ADE is △ABC scaled down by 1/2, and DE ∥ BC.
③With a scale factor of 1 : 2, DE = BC/2. Set the value to 0.5 in the picture above and equal-length marks appear on the four pieces, showing this relation.
Using parallels and ratios
📝 Three common uses
①In a trapezium with a line parallel to the bases, set up a proportion from the parallels.
②With a parallel line inside a triangle, use the scale factor of the similar triangles to find an unknown side.
③Given two midpoints, the midpoint theorem gives a length directly as double or half another.
Worked examples
Example 1
In △ABC, DE ∥ BC with AD = 3, DB = 6, AE = 4. Find EC.
1
The parallel line divides two sides of the triangle, so AD : DB = AE : EC.
3 : 6 = 4 : EC
2
In a proportion, the product of the means equals the product of the extremes.
3 × EC = 24 ⇒ EC = 8
▸ EC = 8
In a triangle, a line parallel to one side divides the other two sides in the same ratio.
Example 2
Three parallel lines cut one transversal in ratio 1 : 2. On another the parts are 3 and x. Find x.
1
Parallel lines divide the two lines in the same ratio.
1 : 2 = 3 : x
2
In a proportion, the product of the means equals the product of the extremes.
1 × x = 2 × 3 ⇒ x = 6
▸ x = 6
The ratio of the segments made by parallel lines is the same whichever line you measure along.
Exam checklist
Parallel lines and segment ratios
ℓ₁ ∥ ℓ₂ ∥ ℓ₃ → a : b = c : d
Parallel lines carry ratios, not lengths. That one sentence runs through the whole topic.
Grade-8 school exam type
In △ABC, M and N are the midpoints of AB and AC. If BC = 10, what is MN?
①4
②5
③6
④8
⑤10
▸ ② 5
1
By the midpoint theorem, MN is parallel to BC and half its length.
MN = (1/2) × BC
2
Substitute BC = 10.
= (1/2) × 10 = 5
🎯 Exam points
①When three parallel lines cut two lines, corresponding segments are in the same ratio.
②In a triangle, if DE ∥ BC then AD : DB = AE : EC.
③The midpoint theorem is the most frequently tested: joining two midpoints gives a parallel segment half as long.
④Solve a proportion using the fact that the product of the means equals the product of the extremes (a : b = c : d → ad = bc).
⑤Watch the converse. In a triangle, a segment dividing two sides in the same ratio is parallel to the third side, but three lines that cut two lines in the same ratio need not be parallel.