Worksheet on similar shapes with algebraic expressions for side lengths.

Let’s solve each problem step by step. We’re dealing with similar shapes, which means their corresponding sides are in proportion — that is, they have the same scale factor.

We’ll go one by one from (a) to (o). For each pair:

1. Identify corresponding sides.
2. Set up a ratio (scale factor) using known sides.
3. Use that ratio to set up an equation for the unknown side.
4. Solve for x (or y).

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a) Rectangles

Small rectangle: width = 5, height = 2
Large rectangle: width = 3x + 1, height = 4

Since they’re similar, ratios of corresponding sides must be equal.

Height ratio: large / small = 4 / 2 = 2 → scale factor is 2.

So width should also scale by 2:
3x + 1 = 2 × 5 = 10
→ 3x = 9
→ x = 3

✔ Check: 3(3)+1 = 10, and 10/5 = 2, matches height ratio 4/2=2. Good.

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b) Right triangles

Small triangle: base = 3, height = 2
Large triangle: base = 3x - 3, height = 6

Scale factor from height: 6 / 2 = 3

So base should be: 3 × 3 = 9
→ 3x - 3 = 9
→ 3x = 12
→ x = 4

✔ Check: 3(4)-3 = 9, 9/3=3, matches height ratio 6/2=3. Good.

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c) Triangles

Small triangle: base = 7, side = 3
Large triangle: base = 3x + 3, side = 9

Side ratio: 9 / 3 = 3 → scale factor 3

Base should be: 7 × 3 = 21
→ 3x + 3 = 21
→ 3x = 18
→ x = 6

✔ Check: 3(6)+3=21, 21/7=3, matches side ratio 9/3=3. Good.

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d) Rectangles

Small: width = 8, height = 4
Large: width = 2x + 2, height = 6

Height ratio: 6 / 4 = 1.5 → scale factor 1.5

Width: 8 × 1.5 = 12
→ 2x + 2 = 12
→ 2x = 10
→ x = 5

✔ Check: 2(5)+2=12, 12/8=1.5, matches 6/4=1.5. Good.

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e) Parallelograms

Small: side = 9, bottom = 4x - 2
Large: side = 12, bottom = 8

Wait — let’s match corresponding sides.

Assume the “side” labeled 9 corresponds to 12, and “bottom” 4x-2 corresponds to 8.

Ratio: 12 / 9 = 4/3

So: 8 / (4x - 2) = 4/3? Wait — better to set up proportion:

Corresponding sides: 9 ↔ 12, and (4x - 2) ↔ 8

So:
9 / 12 = (4x - 2) / 8
Simplify left: 3/4 = (4x - 2)/8
Multiply both sides by 8:
6 = 4x - 2
→ 4x = 8
→ x = 2

✔ Check: 4(2)-2=6; 6/8 = 3/4, and 9/12=3/4. Good.

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f) Trapezoids

Small: top = ? (not given), right side = 4, bottom = 8x + 6
Large: right side = 10, bottom = 25

Assuming right sides correspond: 4 ↔ 10 → scale factor = 10/4 = 2.5

Then bottom: (8x + 6) × 2.5 = 25
→ 8x + 6 = 25 / 2.5 = 10
→ 8x = 4
→ x = 0.5

✔ Check: 8(0.5)+6=4+6=10; 10×2.5=25. Good.

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g) Semicircles

Small: diameter = 4, curved part = 15 - 3x
Large: diameter = 16, curved part = 12

Curved part is half circumference → proportional to diameter.

So ratio of diameters: 16 / 4 = 4 → scale factor 4

Then curved parts: (15 - 3x) × 4 = 12? Wait — no.

Actually, if large is bigger, then:

Large curved / Small curved = Large diameter / Small diameter

So: 12 / (15 - 3x) = 16 / 4 = 4
→ 12 = 4 × (15 - 3x)
→ 12 = 60 - 12x
→ 12x = 48
→ x = 4

✔ Check: 15 - 3(4)=3; 12/3=4, 16/4=4. Good.

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h) Trapezoids

Small: top = 1, bottom = 29 - 3x
Large: top = 5, bottom = 25

Top ratio: 5 / 1 = 5 → scale factor 5

Bottom: (29 - 3x) × 5 = 25
→ 29 - 3x = 5
→ 3x = 24
→ x = 8

✔ Check: 29 - 24 = 5; 5×5=25. Good.

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i) Arrows

Small: arrowhead width = 9, shaft width = 6x - 48
Large: arrowhead width = 12, shaft width = 16

Arrowhead ratio: 12 / 9 = 4/3

Shaft: (6x - 48) × (4/3) = 16
Multiply both sides by 3:
4(6x - 48) = 48
→ 24x - 192 = 48
→ 24x = 240
→ x = 10

✔ Check: 6(10)-48=12; 12 × (4/3) = 16. Good.

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j) Triangles

Small: side = 2, base = 4x - 6
Large: side = 3, base = 6

Side ratio: 3 / 2 = 1.5

Base: (4x - 6) × 1.5 = 6
→ 4x - 6 = 4
→ 4x = 10
→ x = 2.5

✔ Check: 4(2.5)-6=10-6=4; 4×1.5=6. Good.

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k) Star-like shapes

Small: point length = 2, other side = 3x
Large: point length = 4, other side = x + 15

Point ratio: 4 / 2 = 2 → scale factor 2

Other side: 3x × 2 = x + 15
→ 6x = x + 15
→ 5x = 15
→ x = 3

✔ Check: 3(3)=9; 9×2=18; x+15=18. Good.

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l) Quadrilaterals

Small: side = 3, adjacent side = 4x
Large: side = 9, adjacent side = 6x + 36

Side ratio: 9 / 3 = 3 → scale factor 3

Adjacent side: 4x × 3 = 6x + 36
→ 12x = 6x + 36
→ 6x = 36
→ x = 6

✔ Check: 4(6)=24; 24×3=72; 6(6)+36=36+36=72. Good.

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m) Irregular quadrilaterals

Small: side = 4, adjacent = x + 1
Large: side = 12, adjacent = 4x - 5

Side ratio: 12 / 4 = 3 → scale factor 3

Adjacent: (x + 1) × 3 = 4x - 5
→ 3x + 3 = 4x - 5
→ 3 + 5 = 4x - 3x
→ x = 8

✔ Check: x+1=9; 9×3=27; 4(8)-5=32-5=27. Good.

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n) Blob shapes

Small: side = 6, adjacent = 4x + 6
Large: side = 9, adjacent = 10x - 5

Side ratio: 9 / 6 = 1.5 → scale factor 1.5

Adjacent: (4x + 6) × 1.5 = 10x - 5
→ 6x + 9 = 10x - 5
→ 9 + 5 = 10x - 6x
→ 14 = 4x
→ x = 3.5

✔ Check: 4(3.5)+6=14+6=20; 20×1.5=30; 10(3.5)-5=35-5=30. Good.

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o) Wavy shapes

Small: side = 2x + 4, adjacent = 3y + 2
Large: side = 82 - 4x, adjacent = 12y + 8

Set up two proportions.

First, side ratio: (82 - 4x) / (2x + 4)

Second, adjacent ratio: (12y + 8) / (3y + 2)

But since shapes are similar, these ratios must be equal.

Also notice: 12y + 8 = 4*(3y + 2) → so adjacent ratio is 4.

Therefore, side ratio must also be 4:

(82 - 4x) / (2x + 4) = 4
→ 82 - 4x = 4*(2x + 4)
→ 82 - 4x = 8x + 16
→ 82 - 16 = 8x + 4x
→ 66 = 12x
→ x = 5.5

Now plug into adjacent to find y? But we don’t need y unless asked. The question says “solve to find x” — but here there’s also y. Looking back at original: it says “solve to find x”, but in (o) there are both x and y.

Wait — re-read instruction: “For each pair of shapes, form an equation and solve to find x.” So probably only x is required, even if y appears.

But in this case, we can solve for x without y because the adjacent sides give us the scale factor directly: 12y+8 over 3y+2 is always 4, as long as 3y+2 ≠ 0.

So yes, scale factor is 4, so we solved x = 5.5.

✔ Check: 2x+4 = 2(5.5)+4=11+4=15; 82-4x=82-22=60; 60/15=4. Good.

And adjacent: 3y+2 and 12y+8 — ratio is 4 regardless of y (as long as defined). So x is determined independently.

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## Final Answers:

a) x = 3
b) x = 4
c) x = 6
d) x = 5
e) x = 2
f) x = 0.5
g) x = 4
h) x = 8
i) x = 10
j) x = 2.5
k) x = 3
l) x = 6
m) x = 8
n) x = 3.5
o) x = 5.5

Final Answer:
a) 3
b) 4
c) 6
d) 5
e) 2
f) 0.5
g) 4
h) 8
i) 10
j) 2.5
k) 3
l) 6
m) 8
n) 3.5
o) 5.5