Examples of similar shapes with mathematical problems involving perimeter, area, and side lengths.

Let’s solve each part step by step.

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Part a) Find the perimeter of rectangle B.

We are told that rectangles A and B are similar. That means their sides are in proportion.

Rectangle A has one side = 10 cm.
Rectangle B has the corresponding side = 20 cm.

So, the scale factor from A to B is:

> 20 ÷ 10 = 2

That means every side of B is twice as long as the corresponding side of A.

Perimeter scales with the same factor as the sides (because perimeter is linear).

Perimeter of A = 32 cm
→ Perimeter of B = 32 × 2 = 64 cm

✔ Check: Let’s verify using area too — we’ll do that in part b.

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Part b) Find the area of rectangle B.

Area scales with the square of the scale factor.

Scale factor = 2 → Area factor = 2² = 4

Area of A = 60 cm²
→ Area of B = 60 × 4 = 240 cm²

✔ Double-check with dimensions:

If rectangle A has one side = 10 cm, and area = 60 cm², then the other side is:

> 60 ÷ 10 = 6 cm

So rectangle A is 10 cm by 6 cm → perimeter = 2×(10+6) = 32 cm ✔️

Then rectangle B, scaled by 2, is 20 cm by 12 cm → perimeter = 2×(20+12) = 64 cm ✔️
Area = 20 × 12 = 240 cm² ✔️

Perfect.

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Part c) Work out the value of x.

Triangles C and D are similar.

Area of C = 12 cm²
Area of D = 48 cm²

Ratio of areas = 48 ÷ 12 = 4

Since they are similar, the ratio of areas = (scale factor)²

So:

> (scale factor)² = 4 → scale factor = √4 = 2

That means triangle D is twice as big as triangle C in linear dimensions.

Base of D = 8 cm
→ Base of C = x cm

Since D is bigger, and scale factor from C to D is 2:

> x × 2 = 8 → x = 8 ÷ 2 = 4

✔ Check: If base of C is 4 cm, and scale factor is 2, then base of D is 8 cm ✔️
Area ratio should be 2² = 4 → 12 × 4 = 48 ✔️

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Final Answer:
a) 64 cm
b) 240 cm²
c) 4