Math worksheet on similar polygons with exercises to find missing sides, scale factors, and solve equations.

Let’s solve each problem step by step. We’re dealing with similar polygons, which means their corresponding sides are in proportion — that is, they have the same shape but different sizes, and you can find missing lengths using ratios.

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Problem 1: Two similar triangles

Left triangle sides: 10, 8, 14
Right triangle sides: ?, 4, 7

We match corresponding sides. The side labeled “4” on the right corresponds to “8” on the left (both are the shorter slanted sides).
The side labeled “7” on the right corresponds to “14” on the left (the base).

So the scale factor from left to right is:

→ 4 ÷ 8 = ½
→ 7 ÷ 14 = ½ → confirms scale factor is ½

Now, the missing side (?) corresponds to the side of length 10 on the left.

So: ? = 10 × ½ = 5

✔ Answer for #1: 5

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Problem 2: Two similar rectangles

Left rectangle: width = ?, height = 10
Right rectangle: width = 15, height = 24

Wait — actually, looking at the diagram:

Left rectangle: vertical side = 10, horizontal side = ?
Right rectangle: vertical side = 24, horizontal side = 15

But wait — the right rectangle has top/bottom = 15, left/right = 24? That would make it taller than wide. But the left one has height 10 and unknown width.

Actually, let’s check correspondence.

In similar rectangles, corresponding sides must be proportional.

Assume the vertical sides correspond: left height = 10, right height = 24 → ratio = 10/24 = 5/12

Then the horizontal sides should follow same ratio: left width / right width = 5/12

Right width = 15 → so left width = (5/12) × 15 = 75/12 = 6.25? Hmm, not nice number.

Wait — maybe we matched wrong.

Look again: Right rectangle has sides 15 (top/bottom) and 24 (left/right). Left rectangle has 10 (left/right) and ? (top/bottom).

If we assume the *heights* correspond: left height = 10, right height = 24 → ratio = 10/24 = 5/12

Then widths: left width / right width = 5/12 → ? / 15 = 5/12 → ? = (5×15)/12 = 75/12 = 25/4 = 6.25 — still messy.

Alternatively, maybe the *widths* correspond? Left width = ?, right width = 15; left height = 10, right height = 24.

Ratio of heights: 10/24 = 5/12 → then widths should be same ratio: ? / 15 = 5/12 → same as above.

Wait — perhaps I misread the diagram. Let me re-express:

Problem 2 shows:

Left rectangle: labeled “?” on left side, “10” on top? Wait no — looking back at original image description:

Actually, in the user's image description:

For problem 2:

Left rectangle: left side = ?, top = 10
Right rectangle: left side = 24, top = 15

Ah! So:

Left: height = ?, width = 10
Right: height = 24, width = 15

Since they are similar, ratios of corresponding sides equal.

So: height_left / height_right = width_left / width_right

→ ? / 24 = 10 / 15

Simplify 10/15 = 2/3

So ? = 24 × (2/3) = 16

✔ Answer for #2: 16

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Problem 3: Two similar parallelograms

Left: sides 3, 6, 3, ? (bottom)
Right: sides 4, 8, 4, 8

Note: In a parallelogram, opposite sides are equal. So left has two sides of 3, one of 6, and bottom unknown — probably also 6? But wait, if similar, proportions matter.

Actually, left parallelogram: left side = 3, top = 6, right side = 3, bottom = ?

Right parallelogram: left = 4, top = 8, right = 4, bottom = 8

So clearly, the right one has all sides doubled compared to... wait, 3→4? Not double.

Check ratio:

Top side: left = 6, right = 8 → ratio = 6/8 = 3/4

Side: left = 3, right = 4 → ratio = 3/4 → consistent!

So scale factor from left to right is 4/3? Or right to left is 3/4?

We want the missing bottom side of left parallelogram.

In right parallelogram, bottom = 8.

Since left is smaller, and ratio of corresponding sides is 3/4 (left:right), then:

left_bottom / right_bottom = 3/4 → ? / 8 = 3/4 → ? = 6

Wait — but top was already 6. And in a parallelogram, opposite sides are equal — so bottom should equal top → 6.

Yes! So even without similarity, in a parallelogram, opposite sides are equal. So if top is 6, bottom must be 6.

But let’s confirm with similarity: right parallelogram has top=8, bottom=8. Left has top=6, so bottom should be 6. Ratio 6:8 = 3:4, matches side ratio 3:4.

✔ Answer for #3: 6

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Problem 4: Two similar quadrilaterals (with right angles — likely rectangles or trapezoids?)

Left figure: left side = 10, right side = 11, bottom = ?, top not given but has right angles at corners.

Right figure: left side = ?, right side = 22, bottom = 16, top = 12

Wait — actually, looking at labels:

Left: left = 10, right = 11, bottom = ?
Right: left = ?, right = 22, bottom = 16, top = 12

And both have right angle marks at bottom-left and bottom-right? Actually, in the image description, it says:

“4” has two figures with right angles marked at three corners? Probably rectangles? But sides are different.

Wait — perhaps they are trapezoids? But with right angles.

Actually, since they are similar, corresponding sides proportional.

Notice: right figure has right side = 22, left figure has right side = 11 → ratio = 11/22 = 1/2

Similarly, right figure bottom = 16, so left figure bottom should be half → 8

Also, right figure top = 12, so left figure top should be 6 — but not asked.

Left figure left side = 10, so right figure left side should be 20 — but not asked.

We need left figure bottom = ?

From above: since right side ratio is 11:22 = 1:2, then bottom should be 16 × (1/2) = 8

✔ Answer for #4: 8

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Problem 5: Find scale factor between two similar parallelograms

Left: sides 30, 30, 35, ? (but actually, it’s a rhombus? No — labeled 30, 30, 35 — probably adjacent sides 30 and 35)

Looking: left parallelogram: two sides 30, two sides 35? Actually, labeled: top=30, left=30, right=35 — inconsistent.

Wait — better: in problem 5:

Left figure: sides labeled 30 (top), 30 (left), 35 (right) — that doesn’t make sense for parallelogram.

Perhaps it’s a kite? But problem says "polygons are similar".

Actually, standard interpretation: for similar polygons, we compare corresponding sides.

Left parallelogram: let’s say sides are 30 and 35 (adjacent)

Right parallelogram: sides 21 and 18

Which corresponds to which?

Probably, the longer side corresponds to longer side.

Left: 35 and 30 → max 35
Right: 21 and 18 → max 21

So 35 corresponds to 21? Then scale factor = 21/35 = 3/5

Check other pair: 30 corresponds to 18? 18/30 = 3/5 → yes!

So scale factor from left to right is 3/5.

But the question says: “Find the scale factor of the two figures”

Usually, scale factor is written as (image)/(preimage) or sometimes specified. Since it doesn't specify direction, but typically we take smaller/larger or as per order.

In the problem, left is first, right is second. So scale factor from first to second is 3/5.

Sometimes they want it as a fraction or decimal.

✔ Answer for #5: 3/5 or 0.6 — but fraction is better.

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Problem 6: Two similar isosceles triangles

Left: sides 21, 21, base not given
Right: sides 14, 14, base 8

Since similar, corresponding sides proportional.

The equal sides correspond: 21 and 14 → ratio = 14/21 = 2/3

Base of left triangle? Not needed. Scale factor from left to right is 2/3.

Question: “Find the scale factor of the two figures”

Again, assuming from first to second: 14/21 = 2/3

✔ Answer for #6: 2/3

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Problem 7: Solve for x — two similar trapezoids

Left trapezoid: top=45, legs=35,35, bottom=40
Right trapezoid: top=2x+4, legs=?, bottom=16

Since similar, corresponding sides proportional.

First, identify correspondence.

Both are isosceles trapezoids (legs equal).

So top corresponds to top, bottom to bottom, legs to legs.

So ratio of bottoms: 40 / 16 = 5/2? Wait — left bottom=40, right bottom=16 → so if left is larger, ratio left:right = 40:16 = 5:2

Then tops should be same ratio: left top / right top = 5/2

Left top = 45, right top = 2x+4

So: 45 / (2x+4) = 5/2

Cross-multiply: 45 * 2 = 5 * (2x + 4)

90 = 10x + 20

90 - 20 = 10x

70 = 10x

x = 7

Check: 2x+4 = 14+4=18

Ratio tops: 45/18 = 5/2? 45÷9=5, 18÷9=2 → yes 5/2

Bottoms: 40/16 = 5/2 → good.

✔ Answer for #7: 7

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Problem 8: Two similar rectangles

Left: height=3, width=4
Right: height=6, width=x

Similar, so ratios equal.

Height ratio: 6/3 = 2

So width should also scale by 2: x = 4 * 2 = 8

Or set up proportion: 3/6 = 4/x → 1/2 = 4/x → x=8

✔ Answer for #8: 8

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## Final Answers:

1. 5
2. 16
3. 6
4. 8
5. 3/5
6. 2/3
7. 7
8. 8