A right triangle has sides 3, 4, and 5. A smaller triangle is drawn inside it with its vertices on the sides of the larger triangle, such that it is similar to the larger triangle. If its perimeter is 6, what is its area?

Correct — A. Perimeters of similar triangles scale like their sides, and areas scale as the square.

Large perimeter: 3 + 4 + 5 = 12
Scale factor: 6⁄12 = 1⁄2
Large area (right angle between 3 and 4): ½ × 3 × 4 = 6

Area ratio: (1⁄2)² = 1⁄4
Small area: 6 × 1⁄4 = 1.5 → option (a)

Why the others are wrong

• (b)
  2.4 — 2.4 is the large triangle's altitude from the right angle to the hypotenuse (3 × 4 ⁄ 5). It is a length, not the small triangle's area.
• (c)
  3 — 3 scales the area by the linear factor 1⁄2 instead of its square. Areas shrink by (1⁄2)² = 1⁄4, so 6 becomes 1.5.
• (d)
  6 — 6 is the large triangle's own area, ½ × 3 × 4. The smaller triangle, with half the perimeter, has a quarter of that.

If two triangles are similar with side ratio k, every length scales by k: sides, perimeters, altitudes and medians. Area scales by k².

Here k = 1⁄2, so the small triangle has sides 1.5, 2 and 2.5 and area ½ × 1.5 × 2 = 1.5, the same answer by a second route.

The detail about vertices on the sides only places the triangle. The similarity and the perimeter fix its size.

Key facts

• Similar triangles with side ratio k have perimeter ratio k and area ratio k².
• A 3-4-5 triangle is right-angled (3² + 4² = 5²), with area 6.
• The triangle joining the midpoints of the sides is similar to the original with ratio 1⁄2.

Common traps

• Scaling the area by 1⁄2 and answering 3. The perimeter ratio is linear, the area ratio is its square.
• Answering 6, the area of the given 3-4-5 triangle, instead of the inner one.

No directly related past PYQ was found.