Similar triangles · Ages 14–15
Parallel lines and similar triangles
A shape: AD goes with AB (not DB); hourglass: AB goes with CD.
How to do it
Method
A shape: AD goes with AB (not DB); hourglass: AB goes with CD
AD = 2, DB = 3, BC = 10: DE = 10 × 25 = 4
Another way
Write a proportion
DE10 = 25, 5 × DE = 20, DE = 4
In full
Parallel lines often make similar triangles. Shape one (like a letter A): in △ABC, D is on AB and E on AC, with DE // BC. ∠ADE = ∠ABC (corr. ∠s, DE // BC), and ∠A is common, so △ADE ~ △ABC (AAA). Corresponding sides: AD with AB, AE with AC, DE with BC. The commonest slip is to pair AD with DB. AD = 2 and DB = 3: AB = 2 + 3 = 5, so DEBC = ADAB = 25, not 23. Shape two (like an hourglass): AC and BD cross at E, with AB // DC. ∠BAE = ∠DCE (alt. ∠s, AB // DC) and ∠AEB = ∠CED (vert. opp. ∠s), so △ABE ~ △CDE (AAA): AB with CD, AE with CE, BE with DE. One way: the scale first, then multiply. Another: a proportion, cross-multiplied. To check: the big triangle's sides must be longer than the small one's partners.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, DE // BC. x =
Show the answer and working
Answer: 4 cm
△ADE ~ △ABC (AAA), and AB = 3 + 6 = 9 cm: x = 12 × 39 = 4.
Example 2
In the picture, DE // BC. How long is DE?
- A. 10 cm
- B. 36 cm
- C. 3 cm
- D. 9 cm
Choose the right answer
Show the answer and working
Answer: D. 9 cm
△ADE ~ △ABC: DEBC = ADAB = 68, so DE = 12 × 68 = 9 cm.
Example 3
In the picture, AB // DC, AC and BD cross at E, and EC is 5 cm longer than AE. x =
Show the answer and working
Answer: 6 cm
AECE = ABDC: xx + 5 = 611, so 11x = 6x + 6 × 5 and x = 6. Check: 6 : 11 = 6 : 11.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In the picture, AB // DC, and AC and BD cross at E. x =
Answer:
2.In the picture, DE // BC. x =
Answer:
3.In the picture, AB // DC, and AC and BD cross at E. x = , y =
Fill in each blank
Answer:
4.In the picture, DE // BC. How long is DE?
- A. 7.5 cm
- B. 4.5 cm
- C. 9 cm
- D. 6.75 cm
Choose the right answer
Answer:
5.In the picture, DE // BC. x =
Answer:
All the answers
- 1. 9 cm
- 2. 5 cm
- 3. 8, 14
- 4. B. 4.5 cm
- 5. 4 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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