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Parallel lines and proportional parts

الأسبوع 9 · المستقيمات المتوازية والأجزاء المتناسبة: نظرية التناسب في المثلث وعكسها ونظرية تاليس · 3 دروس
3

Thales' general theorem: several parallel lines cut two transversals in proportional parts, the equal-parts case and applications

Mathematics Sec 1 Term 1, Unit 3, Lesson 1
GoalUse Thales’ theorem to find the lengths cut on a transversal by parallel lines.
New words
transversal · a line that crosses two or more other linesThales’ theorem · parallel lines cut every transversal in proportional parts
Explanation

Thales’ theorem: when several parallel lines cut two transversals, the parts on one transversal are proportional to the matching parts on the other: AB/BC = PQ/QR, and in the same way AB : BC : CD = PQ : QR : RS. It extends the triangle theorem: draw through A a line parallel to the second transversal and the parallel lines cut a triangle. A special case: if the parts on one transversal are equal, the parts on every transversal are equal too (this is how we divide a segment into equal parts with a ruler). Real uses: finding the length of a sloping beam between vertical posts, or the distance between two street junctions. Keep the matching order: the part between the first two lines goes with the part between the same two lines on the other transversal. Convert all the lengths to the same unit first.

694xABCPQR
Worked examples
Three parallel lines cut two transversals. On the first, AB = 6 and BC = 9. On the second, PQ = 4. Then AB/BC = PQ/QR, so 6/9 = 4/QR and QR = 6.
Three parallel streets cross two avenues. On the first avenue the blocks are 200 m and 300 m long. On the second avenue the first block is 240 m. Then 200/300 = 240/x, so x = 360 m.
Equal parts. Parallel lines cut two transversals. On the first the three parts are equal. On the second the first part is 3x − 4 and the second part is x + 6. Equal parts give 3x − 4 = x + 6, so 2x = 10, x = 5 and each part is 11.
Class activity

Divide a line. Draw a segment 10 cm long. Using a ruler and a set square, divide it into 5 equal parts by drawing 5 equal steps on a sloping line and joining the ends with parallel lines.

Practice
1
Three parallel lines cut two transversals. On the first AB = 5 and BC = 7; on the second PQ = 10. Find QR and PR.
2
A sloping beam rests on three vertical posts. The posts are 1.5 m and 2.5 m apart along the ground, and the beam between the first and second post is 1.8 m long. How long is the beam between the second and third post?
3
Four equal parts are marked on one transversal by parallel lines. On another transversal the total length cut is 24 cm. Find each part.
4
On one transversal AB : BC = 3 : 5. On the other, PQ = 4.5 cm. Find QR.
5
Why can we not use Thales’ theorem if the lines are not parallel?