The ratios of the perimeters of two similar triangles are expressed as of 3 : 5. Together, their areas total 34 cm². What is the area of the larger triangle?

Correct — C. In similar triangles the perimeters go as the sides, and the areas go as the squares of the sides.

Side ratio = perimeter ratio = 3 : 5
Square it for the area ratio: 3² : 5² = 9 : 25

Add the parts: 9 + 25 = 34
Value of one part: 34 cm² ÷ 34 = 1 cm²
Larger triangle: 25 × 1 = 25 cm² → option (c)

Why the others are wrong

• (a)
  9 cm² — 9 cm² is the smaller triangle's area, the 9 parts of 9 : 25. The question asks for the larger one.
• (b)
  15 cm² — 15 cm² would leave 34 − 15 = 19 cm² for the other triangle, and 15 : 19 is not the 9 : 25 area ratio that sides in 3 : 5 require.
• (d)
  35 cm² — 35 cm² is more than the 34 cm² total of both triangles together, so neither triangle can have that area.

For similar figures with scale factor k, every length (side, perimeter, altitude, median) scales by k, and every area scales by k².

Here k = 5⁄3, so the larger area is 25⁄9 of the smaller. Splitting the 34 cm² total in the ratio 9 : 25 gives 9 cm² and 25 cm².

Key facts

• In similar triangles, the ratio of perimeters equals the ratio of corresponding sides.
• The ratio of areas is the square of the ratio of corresponding sides.
• Sides in 3 : 5 give areas in 9 : 25.

Common traps

• Splitting 34 cm² in the perimeter ratio 3 : 5, which gives 21.25 cm², instead of the area ratio 9 : 25.
• Answering with the smaller triangle's 9 cm² when the question asks for the larger.

No directly related past PYQ was found.