The Size of the Scale Factor - Free Printable

Let’s work through this step by step.

We’re told that triangle ABC is scaled to make triangle DEF using a scale factor of 3/2. That means every side in triangle DEF is 3/2 times the length of the matching side in triangle ABC.

Looking at triangle ABC:
- Side AB = 4
- Side BC = 3
- Side AC = 5

Now, let’s multiply each of these by 3/2 to see if we get the sides of triangle DEF:

→ AB (4) × 3/2 = (4 × 3)/2 = 12/2 = 6 → matches DE in triangle DEF ✔
→ BC (3) × 3/2 = (3 × 3)/2 = 9/2 = 4.5 → matches EF in triangle DEF ✔
→ AC (5) × 3/2 = (5 × 3)/2 = 15/2 = 7.5 → matches DF in triangle DEF ✔

So yes — multiplying each side of ABC by 3/2 gives us the sides of DEF.

Now, what if we go backwards? What if we start with triangle DEF and want to get back to triangle ABC?

We need to “undo” the scaling. Since we multiplied by 3/2 to go from ABC to DEF, we now divide by 3/2 — or equivalently, multiply by its reciprocal: 2/3.

Let’s test it:

→ DE (6) × 2/3 = (6 × 2)/3 = 12/3 = 4 → matches AB ✔
→ EF (4.5) × 2/3 = (4.5 × 2)/3 = 9/3 = 3 → matches BC ✔
→ DF (7.5) × 2/3 = (7.5 × 2)/3 = 15/3 = 5 → matches AC ✔

Perfect! So going from DEF back to ABC uses the reciprocal scale factor: 2/3.

This shows that scaling can be reversed — just use the reciprocal of the original scale factor.

The example at the bottom says: If you scale Figure A by 4 to get Figure B, then to get back to Figure A from Figure B, you scale by 1/4 — which is the reciprocal of 4.

Same idea here: 3/2 and 2/3 are reciprocals. Multiply them together: (3/2) × (2/3) = 6/6 = 1 — which means you’re back to the original size.

Final Answer:
Triangles ABC and DEF are scaled copies of each other. Scaling from ABC to DEF uses a scale factor of 3/2. Scaling back from DEF to ABC uses the reciprocal scale factor, 2/3. This works because multiplying by a number and then by its reciprocal returns you to the original size.