Worksheet on similar shapes with algebraic expressions to find the scale factor and solve for x.

Let’s solve each problem one by one. Since all shapes are similar, their corresponding sides are in proportion. We’ll find the scale factor from the known sides, then set up an equation to solve for x (or y).

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

Scale factor from small to large: height goes from 2 → 4 → scale factor = 2
So width should also multiply by 2:
5 × 2 = 3x + 1
→ 10 = 3x + 1
→ 9 = 3x
→ x = 3

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

Height ratio: 2 → 6 → scale factor = 3
Base: 3 × 3 = 3x - 3
→ 9 = 3x - 3
→ 12 = 3x
→ x = 4

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c)
Small triangle: top side = 3, bottom side = 7
Large triangle: top side = 9, bottom side = 3x + 3

Top side ratio: 3 → 9 → scale factor = 3
Bottom side: 7 × 3 = 3x + 3
→ 21 = 3x + 3
→ 18 = 3x
→ x = 6

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d)
Small rectangle: length = 8, width = 4
Large rectangle: length = 2x + 2, width = 6

Width ratio: 4 → 6 → scale factor = 6/4 = 3/2
Length: 8 × (3/2) = 2x + 2
→ 12 = 2x + 2
→ 10 = 2x
→ x = 5

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e)
Small parallelogram: left side = 9, bottom = 4x - 2
Large parallelogram: left side = 12, bottom = 8

Left side ratio: 9 → 12 → scale factor = 12/9 = 4/3
Bottom: (4x - 2) × (4/3) = 8? Wait — actually, we need to match corresponding sides.

Actually, better to set up proportion directly:

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

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f)
Small trapezoid: right side = 4, bottom = 8x + 6
Large trapezoid: right side = 10, bottom = 25

Right side ratio: 4 → 10 → scale factor = 10/4 = 5/2
Bottom: (8x + 6) × (5/2) = 25? No — wait, if small to large is scale factor 5/2, then:

(8x + 6) × (5/2) = 25 → too big. Actually, let's do proportion:

Small right / Large right = Small bottom / Large bottom
→ 4 / 10 = (8x + 6) / 25
→ 2/5 = (8x + 6)/25
Multiply both sides by 25:
→ 10 = 8x + 6
→ 4 = 8x
→ x = 0.5

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g)
Small semicircle: diameter = 4, curved part labeled 15 - 3x? Wait — looking at diagram:

Actually, it seems like the straight edge (diameter) corresponds:
Small: diameter = 4, “top” label = 15 - 3x
Large: diameter = 16, “top” label = 12

Wait — that doesn’t make sense. Let me re-read.

Looking again:
Small shape: has a flat side labeled 4, and a curved side labeled 15 - 3x
Large shape: flat side labeled 16, curved side labeled 12

But since they’re similar, ratios of corresponding parts must be equal.

Assume flat sides correspond: 4 → 16 → scale factor = 4
Then curved side: (15 - 3x) × 4 = 12? That would mean 15 - 3x = 3 → x=4, but let’s check.

Alternatively, maybe the labels are swapped? Or perhaps the "curved" part is not the perimeter but something else? This is ambiguous.

Wait — perhaps the numbers on the curves are lengths of those arcs? But without more info, maybe it’s meant to be proportional as drawn.

Another interpretation: Maybe the number next to the curve is the length of that arc, and since shapes are similar, arc lengths scale with linear dimensions.

So if diameters are 4 and 16 → scale factor 4
Then arc length should scale by 4: so small arc × 4 = large arc
→ (15 - 3x) × 4 = 12
→ 60 - 12x = 12
→ -12x = -48
→ x = 4

That works. So we’ll go with that.

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h)
Small trapezoid: left side = 1, bottom = 29 - 3x
Large trapezoid: left side = 5, bottom = 25

Left side ratio: 1 → 5 → scale factor = 5
Bottom: (29 - 3x) × 5 = 25? No — that would give 29 - 3x = 5 → x=8, but let’s use proportion:

Small left / Large left = Small bottom / Large bottom
→ 1 / 5 = (29 - 3x) / 25
Multiply both sides by 25:
→ 5 = 29 - 3x
→ 3x = 24
→ x = 8

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i)
Small arrow: vertical stem = 9, horizontal bar = 6x - 48
Large arrow: vertical stem = 12, horizontal bar = 16

Stem ratio: 9 → 12 → scale factor = 12/9 = 4/3
Bar: (6x - 48) × (4/3) = 16? Or set proportion:

Small stem / Large stem = Small bar / Large bar
→ 9 / 12 = (6x - 48) / 16
→ 3/4 = (6x - 48)/16
Multiply both sides by 16:
→ 12 = 6x - 48
→ 60 = 6x
→ x = 10

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j)
Small triangle: two sides given: 2 and ? Wait — diagram shows:

Small triangle: sides 2 and ? Actually, looks like:

Small triangle: one side = 2, another side = ? Not clear. Wait — labels:

Small triangle: side labeled 2, and side labeled 4x - 6?
Large triangle: side labeled 3, and side labeled 6? And another side labeled 3?

Actually, looking carefully:

Small triangle: sides 2 and 4x - 6
Large triangle: sides 3 and 6? But also has a side labeled 3 — probably equilateral? No.

Better: assume corresponding sides.

If small triangle has sides 2 and 4x-6, large has 3 and 6 — likely 2 corresponds to 3, and 4x-6 corresponds to 6.

So ratio: 2/3 = (4x - 6)/6
Cross multiply: 2×6 = 3×(4x - 6)
→ 12 = 12x - 18
→ 30 = 12x
→ x = 30/12 = 2.5 → x = 2.5

Or fraction: x = 5/2

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k)
Small star-like shape: sides 2 and 3x
Large: sides 4 and x + 15

Assume 2 corresponds to 4 → scale factor 2
Then 3x corresponds to x + 15 → 3x × 2 = x + 15? No — if small to large is scale factor 2, then:

Small side × 2 = large side
So 3x × 2 = x + 15 → 6x = x + 15 → 5x = 15 → x=3

Check other pair: 2 × 2 = 4 → matches. Good.

→ x = 3

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l)
Small quadrilateral: sides 3 and 4x
Large: sides 9 and 6x + 36

Ratio: 3 → 9 → scale factor 3
So 4x × 3 = 6x + 36
→ 12x = 6x + 36
→ 6x = 36
→ x = 6

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m)
Small irregular shape: sides 4 and x+1
Large: sides 12 and 4x - 5

Ratio: 4 → 12 → scale factor 3
So (x + 1) × 3 = 4x - 5
→ 3x + 3 = 4x - 5
→ 3 + 5 = 4x - 3x
→ x = 8

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n)
Small shape: sides 6 and 4x + 6
Large: sides 9 and 10x - 5

Ratio: 6 → 9 → scale factor 1.5 or 3/2
So (4x + 6) × (3/2) = 10x - 5
Multiply both sides by 2:
→ 3(4x + 6) = 2(10x - 5)
→ 12x + 18 = 20x - 10
→ 18 + 10 = 20x - 12x
→ 28 = 8x
→ x = 28/8 = 3.5 → x = 7/2

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o)
This one has y, not x. Two shapes.

Small: sides 2x + 4 and 3y + 2
Large: sides 82 - 4x and 12y + 8

Since similar, ratios of corresponding sides equal.

Assume 2x+4 corresponds to 82-4x, and 3y+2 corresponds to 12y+8.

First, set ratio for x-sides:

(2x + 4) / (82 - 4x) = same as y-sides ratio: (3y + 2)/(12y + 8)

But notice: 12y + 8 = 4*(3y + 2) → so ratio for y-sides is 1/4

Therefore, x-sides must also have ratio 1/4:

(2x + 4) / (82 - 4x) = 1/4
Cross multiply:
4(2x + 4) = 1(82 - 4x)
→ 8x + 16 = 82 - 4x
→ 8x + 4x = 82 - 16
→ 12x = 66
→ x = 66/12 = 11/2 = 5.5

Now for y: since ratio is 1/4, and large side is 4 times small side, which matches: 12y+8 = 4*(3y+2) → always true. So no new info. But we can pick any value? Wait — actually, the ratio is fixed, so y can be anything? But that can't be.

Wait — perhaps we need to use the fact that the scale factor is consistent. From x, we found scale factor is 4 (since small/large = 1/4). So for y, same thing: small side * 4 = large side → which is already satisfied identically. So y is free? But that doesn’t make sense for a homework problem.

Perhaps I misassigned correspondence.

Alternative: maybe 2x+4 corresponds to 12y+8, and 3y+2 corresponds to 82-4x? Unlikely.

Or perhaps the shapes are oriented differently.

Another idea: maybe the scale factor is the same for both pairs, so:

(2x + 4) / (82 - 4x) = (3y + 2) / (12y + 8)

But as noted, (3y+2)/(12y+8) = 1/4, so:

(2x+4)/(82-4x) = 1/4 → which gives x=5.5 as above.

Then for y, since the ratio is 1/4, and it's automatically satisfied, perhaps y can be any number? But that seems odd.

Wait — look at the large shape: side labeled 12y + 8, and small has 3y + 2. If scale factor is 4, then 4*(3y+2) = 12y + 8 → identity. So no constraint on y. But the problem asks to solve for x and y? The instruction says "solve to find x", but here there's a y.

Looking back at original problem: "solve to find x" — but in o), variables are x and y. Probably typo, or we need to find both.

But with current setup, y is not determined. Unless...

Perhaps the correspondence is different. Suppose:

Small side A: 2x+4 corresponds to large side B: 12y+8
Small side C: 3y+2 corresponds to large side D: 82-4x

Then:

(2x+4)/(12y+8) = (3y+2)/(82-4x)

And since shapes are similar, this ratio should be constant, say k.

But we have two equations? Too many variables.

Notice that 12y+8 = 4*(3y+2), so if we let s = 3y+2, then large corresponding side is 4s.

Similarly, suppose the other pair: let t = 2x+4, then large side is 82-4x.

But 82-4x = 82 - 2*(2x) = 82 - 2*(t - 4) = 82 - 2t + 8 = 90 - 2t

Set ratio: t / (90 - 2t) = s / (4s) = 1/4

Same as before: t/(90-2t) = 1/4 → 4t = 90 - 2t → 6t=90 → t=15 → 2x+4=15 → 2x=11 → x=5.5

Then for y, since s = 3y+2, and ratio is 1/4, but no further constraint, so y can be anything? But that can't be.

Unless the problem expects us to realize that the scale factor is 4, and thus for the y-part, it's consistent for all y, but perhaps we need to report x only? But the problem says "solve to find x", and in o) there is y.

Looking at the image description: in o), it's labeled with x and y, and the instruction is "solve to find x", so probably y is a distractor or we need to express in terms, but unlikely.

Another possibility: perhaps the two expressions with y are corresponding, and we can set their ratio equal to the x-ratio.

We have from x: scale factor large/small = 4 (since 82-4x divided by 2x+4 = 4 when x=5.5: 82-22=60, 2*5.5+4=15, 60/15=4)

For y: large side / small side = (12y+8)/(3y+2) = 4, as established.

So it's always 4, regardless of y. So y is not constrained. But that means the problem might have a mistake, or perhaps we are to find x only, and y is irrelevant.

But the problem says "for each pair... solve to find x", so in o), even though there's y, we only solve for x, and y can be any number, but that seems odd.

Perhaps in the diagram, the sides are paired differently. Let me try assuming:

Small: 2x+4 and 3y+2
Large: 82-4x and 12y+8

And suppose 2x+4 corresponds to 12y+8, and 3y+2 corresponds to 82-4x.

Then:

(2x+4)/(12y+8) = (3y+2)/(82-4x)

Let r = this ratio.

Also, since similar, the ratios should be equal, but we have one equation with two variables.

Notice that 12y+8 = 4(3y+2), so let u = 3y+2, then 12y+8 = 4u

Equation: (2x+4)/(4u) = u/(82-4x)

So (2x+4)(82-4x) = 4u^2

But u = 3y+2, still messy.

From earlier approach, if we assume the natural correspondence (first with first, second with second), we get x=5.5, and y arbitrary. But perhaps the problem intends for us to use the fact that the scale factor is the same, and since for y it's automatically satisfied, we just report x.

Maybe in the context, y is to be found similarly, but it's underdetermined.

Another thought: perhaps "3y+2" and "12y+8" are not both sides, but one is a side and the other is something else? Unlikely.

Or perhaps it's a typo, and it's supposed to be only x.

Given the pattern, and that in all other problems we solve for x, and here x is solvable, while y is not constrained, I think we should report x=5.5, and note that y can be any number, but since the problem likely expects a numerical answer, and in the calculation for x it's determined, we'll go with x=11/2.

But let's double-check with actual values.

If x=5.5, small side 2x+4=11+4=15, large side 82-4*5.5=82-22=60, ratio 60/15=4.

For y, small side 3y+2, large side 12y+8=4*(3y+2), so ratio is always 4, for any y. So indeed, y is free. But since the problem says "solve to find x", we provide x.

So x = 5.5 or 11/2

But to be precise, let's write as fraction.

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Now, compiling all 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 or 5/2
k) x=3
l) x=6
m) x=8
n) x=3.5 or 7/2
o) x=5.5 or 11/2

For f, j, n, o, we have decimals or fractions. Better to use fractions where possible.

f) x=1/2
j) x=5/2
n) x=7/2
o) x=11/2

Now, final answer format: since the user didn't specify which one, but the task is to solve all, probably list all.

But the instruction says "Final Answer: [Provide only the final correct answer]", implying perhaps for the whole set.

Since it's multiple parts, I'll list them.

But to follow strictly, perhaps box all.

Given the complexity, I'll present the answers in order.

Final Answer:
a) 3
b) 4
c) 6
d) 5
e) 2
f) \frac{1}{2}
g) 4
h) 8
i) 10
j) \frac{5}{2}
k) 3
l) 6
m) 8
n) \frac{7}{2}
o) \frac{11}{2}