KA-0164Scaling and similar figures
4 points, difficulty 2 of 3Level 9–10about 120sTwo triangles are similar. The sides of the smaller are 3/5 as long as the matching sides of the larger. The smaller triangle has area 18 square cm. What is the area of the larger triangle?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Area does not scale the same way length does.
2The strategy
If every length is multiplied by k, the area is multiplied by k squared.
3The full solution
The larger sides are 5/3 of the smaller, so its area is (5/3) squared = 25/9 times as big: 18 x 25/9 = 50.
Solution
The reliable way
Under a scale factor k, every length multiplies by k but area multiplies by k squared, because area is a product of two lengths. Going from the smaller triangle to the larger multiplies lengths by 5/3, so it multiplies area by (5/3)^2 = 25/9. The larger area is therefore 18 x 25/9 = 50 square cm. Sanity check the direction: the larger triangle must have the larger area, which rules out anything below 18. The transferable idea: lengths scale by k and areas by k squared, because an area is a product of two lengths.
The elegant way
Areas of similar figures are in the ratio 9 to 25, and 18 is 9 x 2, so the larger is 25 x 2 = 50.
Why this is on the test: Scaling area by the length factor rather than its square is one of the most reliably examined mistakes in geometry.