KA-0087Scaling and similar figures
4 points, difficulty 2 of 3Level 5–6about 90sTwo similar triangles have areas of 9 and 25 square units. A side of the smaller one is 6 units. How long is the matching side of the larger one?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Areas scale by the square of the length factor.
2The strategy
If every length is multiplied by k then every area is multiplied by k squared. Work backwards from the area ratio to find k, and only then apply it to the side.
3The full solution
The area ratio is 25 to 9, so the length ratio is 5 to 3. Then 6 x 5 divided by 3 = 10.
Solution
The reliable way
Lengths scale by k and areas by k squared. Here the areas are in the ratio 25 to 9, so k squared is 25/9 and k is 5/3. Applying that to the side: 6 x 5/3 = 10. Check: the larger triangle's area should be (5/3) squared = 25/9 times the smaller, which it is. The transferable idea: take the square root of an area ratio before applying it to a length.
The elegant way
9 and 25 are 3 squared and 5 squared, so the shapes are in the ratio 3 to 5 and the side goes 6 to 10.
Why this is on the test: Applying an area ratio directly to a length is one of the highest-frequency errors on the whole paper, at every band.