KA-0042Scaling and similar figures
4 points, difficulty 2 of 3Level 5–6about 75sA photograph is enlarged so that each side becomes 3 times as long. How many times bigger is its area?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Try it on a small rectangle first.
2The strategy
Take a rectangle you can compute, triple both sides, and compare the two areas. The relationship you find holds for any shape, not just that rectangle.
3The full solution
A 2 by 5 rectangle has area 10. Tripled it becomes 6 by 15 with area 90, which is 9 times as much. Area scales by the square of the length factor.
Solution
The reliable way
Test it concretely. A 2 by 5 rectangle has area 10. After tripling each side it is 6 by 15, with area 90 - nine times the original. In general, tripling both dimensions multiplies the area by 3 x 3 = 9. The transferable idea: length scales by k, area by k squared, volume by k cubed.
The elegant way
Nine copies of the original photo tile the enlarged one exactly, three across and three down.
Why this is on the test: Applying a length scale factor to an area is one of the highest-frequency errors across the whole exam, at every band.