Similarity & Proportionality — Drills
Similarity & Proportionality has 11 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Aa
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two pairs of congruent angles prove two triangles
- similar
- congruent
- right-angled
- isosceles
Worked answer
similar
The third angle follows, so all three match.
Same shape, any size — similar, not congruent.
Scale
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
△ ABC ∼ △ DEF with AB = 7 and DE = 10.5. Find the scale factor.
Worked answer
1.5
k = 10.57 = 1.5
Problem 2
△ ABC ∼ △ DEF with AB = 10 and DE = 50. Find the scale factor.
Worked answer
5
k = 5010 = 5
Side
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
△ ABC ∼ △ DEF: AB = 9, DE = 13.5, BC = 4. Find EF.
Worked answer
6
k = 1.5
EF = 1.5 × 4 = 6
Problem 2
△ ABC ∼ △ DEF: AB = 2, DE = 8, BC = 9. Find EF.
Worked answer
36
k = 4
EF = 4 × 9 = 36
Prop
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A line parallel to the base cuts one side into 8 (top) and 7 (bottom), and the other into 12 (top) and x (bottom). Find x.
Worked answer
10.5
87 = 12x
8x = 84
x = 10.5
Problem 2
A line parallel to the base cuts one side into 2 (top) and 6 (bottom), and the other into 6 (top) and x (bottom). Find x.
Worked answer
18
26 = 6x
2x = 36
x = 18
Shadow
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A 4 ft person casts a 8 ft shadow. At the same moment a tree casts a 112 ft shadow. Find the tree height.
Worked answer
56
48 = t112
t = 4 × 14 = 56 ft
Problem 2
A 6 ft person casts a 6 ft shadow. At the same moment a tree casts a 36 ft shadow. Find the tree height.
Worked answer
36
66 = t36
t = 6 × 6 = 36 ft
Area
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two similar figures have scale factor 1.5. The smaller has area 36. Find the larger area.
Worked answer
81
k2 = 2.25
36 × 2.25 = 81
Problem 2
Two similar figures have scale factor 4. The smaller has area 10. Find the larger area.
Worked answer
160
k2 = 16
10 × 16 = 160
Backarea
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two similar triangles have areas 9 and 144. Find the scale factor.
Worked answer
4
k2 = 1449 = 16
k = 4
Problem 2
Two similar triangles have areas 7 and 175. Find the scale factor.
Worked answer
5
k2 = 1757 = 25
k = 5
Test
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which similarity test uses two pairs of congruent angles?
Worked answer
AA similarity
Two pairs of congruent angles.
That is AA similarity.
Problem 2
Which similarity test uses two pairs of sides in ratio with the included angles congruent?
Worked answer
SAS similarity
Two pairs of sides in ratio with the included angles congruent.
That is SAS similarity.
Volume
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two similar solids have sides in ratio 1:2. Find the ratio of their volumes.
Worked answer
1:8
k3 = 23 = 8
Ratio 1:8
Problem 2
Two similar solids have sides in ratio 1:3. Find the ratio of their volumes.
Worked answer
1:27
k3 = 33 = 27
Ratio 1:27
Perimeter
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Similar triangles have scale factor 4. The smaller has perimeter 10. Find the larger.
Worked answer
40
10 × 4 = 40
Problem 2
Similar triangles have scale factor 3. The smaller has perimeter 16. Find the larger.
Worked answer
48
16 × 3 = 48
Mean
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the geometric mean of 16 and 16.
Worked answer
16
√16 × 16 = √256
= 16
Problem 2
Find the geometric mean of 25 and 25.
Worked answer
25
√25 × 25 = √625
= 25