Similarity & Proportionality — Practice
Equal angles force proportional sides. That single fact measures buildings, mountains and stars.
What these questions cover
These questions use 21 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Scale factor — The number every length is multiplied by.
- Similar figures — Same shape, different size: equal angles, proportional sides.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Volume — Space inside, in cubic units.
- Base — The number being multiplied. In 53 the base is 5.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Scale — What one grid step is worth — and where the axis starts.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Similar — Same shape, possibly different size. Written △ ABC ∼ △ DEF.
- Parallel — Same slope, different intercept.
- Ray — One endpoint and everything beyond it in one direction. Written AB.
- Prove — The statement you must arrive at. Always the last line.
- Included angle — The angle between two named sides. The A in SAS must be included.
- AA similarity — Two pairs of congruent angles prove two triangles similar.
- SSS similarity — All three pairs of sides in the same ratio.
- SAS similarity — Two pairs of sides in ratio with the included angles congruent.
- Geometric mean — For a, x, b in proportion, x = √ab.
- Pole — The origin, where r = 0.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
Two pairs of congruent angles prove two triangles
- similar
- congruent
- isosceles
- right-angled
Worked answer
similar
The third angle follows, so all three match.
Same shape, any size: similar.
Problem 2
△ ABC ∼ △ DEF with AB = 4 and DE = 10. Find the scale factor.
Worked answer
2.5
k = 104 = 2.5
Problem 3
△ ABC ∼ △ DEF, AB = 4, DE = 10, BC = 6. Find EF.
Worked answer
15
k = 2.5
EF = 2.5 × 6 = 15
Problem 4
A line parallel to the base cuts one side 4 and 6, the other 6 and x. Find x.
Worked answer
9
46 = 6x
4x = 36
x = 9
Problem 5
A 5 ft person casts a 4 ft shadow. A tree casts a 32 ft shadow. Find the tree height.
Worked answer
40
54 = h32
h = 40 ft
Problem 6
Two similar figures have sides in ratio 2:3. Find the ratio of their areas.
Worked answer
4:9
k = 32, so k2 = 94
Ratio 4:9
Problem 7
Similar pentagons have sides 6 and 15. The smaller has area 48. Find the larger area.
Worked answer
300
k = 2.5, k2 = 6.25
48 × 6.25 = 300
Problem 8
Two similar triangles have areas 9 and 81. Find the ratio of their sides.
Worked answer
1:3
k2 = 9
k = 3, so sides are 1:3
Problem 9
In △ ABC, DE ∥ BC with AD = 6, DB = 4, AE = 9. Find EC.
Worked answer
6
64 = 9EC
6 · EC = 36
EC = 6
Problem 10
Two pairs of sides in ratio with the included angles congruent proves
- SAS similarity
- SSS similarity
- AA similarity
- congruence
Worked answer
SAS similarity
Two sides in ratio plus the angle between them.
That is SAS similarity.
Problem 11
Two similar solids have sides in ratio 1:2. Find the ratio of their volumes.
Worked answer
1:8
k3 = 8
Ratio 1:8
Problem 12
Similar triangles have scale factor 3. The smaller has perimeter 14. Find the larger perimeter.
Worked answer
42
14 × 3 = 42
Problem 13
A 6 ft pole casts a 9 ft shadow. At the same time a building casts a 120 ft shadow. Find its height.
Worked answer
80
69 = h120
9h = 720
h = 80 ft
Problem 14
A line parallel to one side cuts AD = x, DB = 8, AE = 6, EC = 12. Find x.
Worked answer
4
x8 = 612
12x = 48
x = 4
Problem 15
A model is built at scale 1:20. The model uses 0.5 m2 of card. How much would the real one need?
Worked answer
200
k2 = 400
0.5 × 400 = 200 m2
Problem 16
Find the geometric mean of 4 and 25.
Worked answer
10
√4 × 25 = √100
= 10