Geometry and proof · Ages 14–15
Similar triangles worksheet
Covers: Corresponding sides of similar triangles, Conditions for similar triangles, Parallel lines and similar triangles, Proving triangles similar, Similar polygons, Areas of similar figures, The Sierpinski triangle
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Similar triangles worksheet
Geometry and proof · Ages 14–15
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.△ABC ~ △PQR, with AB = 6 cm, BC = 10 cm and PQ = 7 cm. QR = cm (as a fraction)
Answer in simplest form
Answer:
2.△ABC ~ △DEF, ∠A = 55° and ∠B = 80°. ∠F =
Answer:
3.In the picture, AB // DC, and AC and BD cross at E. x =
Answer:
4.In the picture, AB // DC, AC and BD cross at E, and EC is 3 cm longer than AE. x =
Answer:
5.In the picture, DE // BC. Find DE: put the steps in order.
- A. DE = 10 × 610 = 6 cm
- B. DEBC = ADAB (corr. sides, ~△s)
- C. ∠A = ∠A (common ∠), ∠ADE = ∠ABC (corr. ∠s, DE // BC), so △ADE ~ △ABC (AAA)
Write the letters in the right order
Answer:
6.In the picture, ABCD ~ PQRS. x =
Answer:
7.An equilateral triangle has sides of 8 cm. Made into a Sierpinski triangle, after step 3 the perimeters of all the small shaded triangles add up to
Answer:
Part B: Choose, or true or false (2 marks each)
8.△ABC ~ △PRQ, with AB = 9 cm, BC = 5 cm, CA = 8 cm and PR = 27 cm. How long is QR?
- A. 24 cm
- B. 23 cm
- C. 30 cm
- D. 15 cm
Choose the right answer
Answer:
9.A triangle with sides 3 cm, 6 cm and 8 cm is similar to one with sides 6 cm, 9 cm and 11 cm.
True or false?
Answer:
10.In the picture, AB // DC, and AC and BD cross at E. In △ABE and △CDE: ∠BAE = ∠DCE ( ? ). Which reason goes in the gap?
- A. corr. ∠s, AB // DC
- B. common ∠
- C. alt. ∠s, AB // DC
- D. vert. opp. ∠s
Choose the right answer
Answer:
11.A rectangle 7 cm long and 3 cm wide is similar to one 14 cm long and 7 cm wide.
True or false?
Answer:
12.Two similar polygons have sides in the ratio 4 : 5. The small one's area is 96 cm². What is the big one's?
- A. 240 cm²
- B. 61.44 cm²
- C. 120 cm²
- D. 150 cm²
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.Two triangular flags have the same shape. The small one's sides are 18 cm, 22 cm and 25 cm, and the big one's shortest side is 27 cm. Scale = , and the big flag's perimeter = cm.
Fill in each blank
Working:
Answer:
14.A triangular stand is made of rods 4 dm, 8 dm and 10 dm long. To make a bigger stand of the same shape, which three rods should be used?
- A. 8 dm, 16 dm, 11 dm
- B. 8 dm, 16 dm, 20 dm
- C. 8 dm, 8 dm, 20 dm
- D. 5 dm, 9 dm, 11 dm
Choose the right answer
Working:
Answer:
15.A tent's cross-section is a triangle ABC with the ground BC 7 m wide. A rope DE is tied to AB and AC parallel to the ground, with AD = 3 m and DB = 4 m. AB = m, and the rope DE = m.
Fill in each blank
Working:
Answer:
16.In the picture, AB // DC, and AC and BD cross at E. Prove ABCD = AECE: put the steps in order.
- A. ∠AEB = ∠CED (vert. opp. ∠s), so △ABE ~ △CDE (AAA)
- B. ∠BAE = ∠DCE (alt. ∠s, AB // DC)
- C. ABCD = AECE (corr. sides, ~△s)
Write the letters in the right order
Working:
Answer:
17.Two TVs have rectangular screens of the same shape. The small screen is 20 cm long and 15 cm wide; the big one is 54 cm wide. How long is the big screen?
Working:
Answer:
18.A craftsman makes a Sierpinski triangle in gold leaf to step 4, one sheet on each small shaded triangle at $3 a sheet. How much is the gold leaf?
Working:
Answer:
Show the answers and working
Similar triangles worksheet: Answers and working
Part A: Working out and filling in
1. 353
QRBC = PQAB: QR10 = 76, so QR = 10 × 7 ÷ 6 = 353.
2. 45°
∠F goes with ∠C = 180° − 55° − 80° = 45° (∠ sum of △; corr. ∠s, ~△s).
3. 10 cm
△ABE ~ △CDE (AAA): CE ÷ AE = 8 ÷ 4 = 2, so x = 5 × 2 = 10 (corr. sides, ~△s).
4. 5 cm
AECE = ABDC: xx + 3 = 58, so 8x = 5x + 5 × 3 and x = 5. Check: 5 : 8 = 5 : 8.
5. C → B → A
Similarity first; only then do the corresponding sides give the proportion, and last AB = 10 cm goes in.
6. 85°
x is ∠S, which goes with ∠D = 360° − 95° − 90° − 90° = 85°: similar figures' angles are equal, never scaled.
7. 81 cm
Each small triangle has sides 8 ÷ 23 = 1 cm and a perimeter of 3 × 1 cm, and there are 33: 81 cm. Or times 32 each step.
Part B: Choose, or true or false
8. D. 15 cm
A goes with P, B with R, C with Q. The scale is PR ÷ AB = 27 ÷ 9 = 3, and QR goes with BC: 5 × 3 = 15 cm.
9. False
Divide pair by pair, short to long: 63, 96, 118. Only three equal ratios make the sides proportional; adding the same number to each side doesn't.
10. C. alt. ∠s, AB // DC
AC cuts the parallels AB and DC, and ∠BAE and ∠DCE are on either side of it: alt. ∠s, AB // DC.
11. False
Rectangles' angles are all 90°, so look at the sides: similar only if 147 and 73 are equal.
12. D. 150 cm²
Areas are 42 : 52, so the big one = 96 × 52 ÷ 42 = 150 cm².
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 1.5, 97.5
Shortest with shortest: 27 ÷ 18 = 1.5; the perimeter is multiplied by 1.5 too: (18 + 22 + 25) × 1.5 = 97.5 cm.
14. B. 8 dm, 16 dm, 20 dm
Every rod times 2: 84 = 168 = 2010 = 2, 3 sides proportional.
15. 7, 3
AB = 3 + 4 = 7 m; DE // BC, so △ADE ~ △ABC: DE = 7 × 3 ÷ 7 = 3 m.
16. B → A → C
Two pairs of equal angles first (alternate, vertically opposite), then similarity, and only then the corresponding sides.
17. 72 cm
Length : width = 20 : 15 = 4 : 3, so the big screen's length = 54 × 4 ÷ 3 = 72 cm.
18. 243 dollars
34 small triangles at $3 each: 243 dollars.
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