Final answer:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides (e.g., Area(ΔABC)/Area(ΔPQR) = AB²/PQ²), because side lengths and heights are proportional in similar triangles.
Step-by-step explanation:
To prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides, begin by considering two similar triangles ΔABC and ΔPQR. Because the triangles are similar, their sides are proportional, meaning if AB corresponds to PQ, then AB/PQ = BC/QR = AC/PR.
Now, the area of a triangle can be expressed as 1/2 × base × height. When two triangles are similar, the heights are proportional to their corresponding sides. Thus, the ratio of the areas of ΔABC and ΔPQR can be expressed as the ratio of their bases times the ratio of their heights.
Area(ΔABC) / Area(ΔPQR) = (1/2 × AB × height of ΔABC) / (1/2 × PQ × height of ΔPQR) = (AB/PQ) × (height of ΔABC/height of ΔPQR).
Since the heights are proportional to the sides, we have height of ΔABC/height of ΔPQR = AB/PQ, and thus:
Area(ΔABC) / Area(ΔPQR) = (AB/PQ)^2
By applying this logic to the other pairs of corresponding sides, the following correct options can be deduced based on the corresponding sides discussed:
- Area(ΔABC)/Area(ΔPQR) = AB2/PQ2
- Area(ΔABC)/Area(ΔPQR) = BC2/QR2
- Area(ΔABC)/Area(ΔPQR) = AC2/PR2