A is like B, B is like C

The idea

If one polygon is similar to a second polygon and the second is similar to a third, then the first polygon is similar to the third.

Three photographs

Photo A is small. Photo B is a zoomed copy of A. Photo C is a zoomed copy of B.

So A is similar to B, and B is similar to C.

Is C a zoomed copy of A too?

• Yes. A is similar to C.
• No. C is only similar to B.
• Only if A and C have the same size.

Find each zoom

The blue house is zoomed to fit the orange one. Pick a pair and turn the zoom dial until it fits exactly. Find all three zooms.

Similarity passes along

If polygon 1 is similar to polygon 2, and polygon 2 is similar to polygon 3, then polygon 1 is similar to polygon 3.

The zooms multiply: A to B is ×1.5 and B to C is ×2, so A to C is 1.5 × 2 = ×3. The corners stay equal at every step.

Notes

If polygon 1 is similar to polygon 2 and polygon 2 is similar to polygon 3, then polygon 1 is similar to polygon 3.

Check yourself

Triangle P is similar to triangle Q, and triangle Q is similar to triangle R. What can you say about P and R?

A is zoomed by ×2 to give B. Then B is zoomed by ×5 to give C. By what number is A zoomed to give C?

Answer: 10

2 × 5 = 10, so A fits C with a zoom of ×10.

Square S₁ has side 2 cm. Zooming it by ×3 gives square S₂. Zooming S₂ by ×2 gives square S₃. What is the side of S₃, in cm?

Answer: 12

S₂ has side 2 × 3 = 6 cm. S₃ has side 6 × 2 = 12 cm. S₃ is S₁ zoomed by ×6.

Photos A and C are both similar to the same photo B. What about A and C?

• P and R are congruent.. Not necessarily. P could be much smaller than R. They are similar, but congruent only if the zooms cancel to ×1.
• Nothing. P and R may have different shapes.. P has the shape of Q, and Q has the shape of R. So P has the shape of R.
• P and R are similar. — correct. Yes! Zoom P to fit Q, then zoom Q to fit R.
• A and C are similar. Zoom A to fit B, then zoom B to fit C. — correct. Yes! A fits B and B fits C, so A is similar to C.
• A and C may have different shapes.. A has the shape of B, and C has the shape of B. So A and C have the same shape.
• A and C must be the same size.. Similar figures can have different sizes. A and C are similar, not necessarily congruent.