The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which is equal in area to the triangle. It has a width of 10 units, then the perimeter of this rectangle is :

Correct — option (2), 26 units.

Step 1 — test whether the triangle is right-angled, using the longest side.
5² + 12² = 25 + 144 = 169
13² = 169
The two agree, so the angle between the sides 5 and 12 is a right angle.

Step 2 — area of the triangle, with the two shorter sides as base and height.
Area = ½ × 5 × 12 = 30 square units

Step 3 — the rectangle has the same area and a width of 10.
Other side = 30 ÷ 10 = 3 units

Step 4 — perimeter of the rectangle.
Perimeter = 2 × (10 + 3) = 26 units

Cross-check with Heron's formula, which does not need the right angle:
s = (5 + 12 + 13) ÷ 2 = 15
Area = √(15 × 10 × 3 × 2) = √900 = 30 square units

The idea to remember: 5, 12, 13 is a Pythagorean triple, so the sides 5 and 12 are the base and height.

Why the others are wrong

• (1)
  30 units — 30 is a number from the triangle, not the rectangle. It is the triangle's area (30 square units) and also the triangle's perimeter (5 + 12 + 13 = 30 units).

  The stem asks for the rectangle's perimeter. With sides 10 and 3, that is 2 × 13 = 26 units.

• (3)
  13 units — 13 is length + width, 10 + 3, which is half the rectangle's perimeter. A perimeter runs round all four sides, so the sum is doubled: 26 units.

  13 also appears in the stem as the triangle's longest side, which plays no part in the rectangle's perimeter.

• (4)
  40 units — A perimeter of 40 with one side 10 needs the other side to be 10 as well. That is a 10 × 10 square with an area of 100 square units.

  The triangle's area is 30 square units, so the rectangle's other side is 3, not 10, and the perimeter is 26 units.

The converse of Pythagoras' theorem: if the sides of a triangle satisfy a² + b² = c², the angle opposite c is a right angle. Whole-number sets like this are Pythagorean triples, such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25.

In a right triangle the two shorter sides are perpendicular, so area = ½ × one leg × the other.

For any triangle with sides a, b, c, Heron's formula gives area = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.

A rectangle's area is length × width, and its perimeter is 2 × (length + width).

Key facts

• If a² + b² = c² for a triangle's sides, the angle opposite side c is a right angle.
• Pythagorean triples include 3-4-5, 5-12-13, 8-15-17 and 7-24-25; multiples such as 6-8-10 also satisfy a² + b² = c².
• Area of a right triangle = ½ × the product of the two perpendicular sides; for 5-12-13 it is 30 square units.
• Heron's formula: area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2.
• Rectangle: area = length × width and perimeter = 2 × (length + width); equal areas can have different perimeters.

Common traps

• Forgetting the ½ and taking 5 × 12 = 60 as the area. That gives the other side as 6 and a perimeter of 32, which is not an option.
• Stopping at length + width = 13, which is half the perimeter.
• Using the longest side, 13, as base or height. It is the hypotenuse and is not perpendicular to either of the other sides.

UnlockIAS will link similar questions from RAS Pre 2023 here once that paper is published on this site.

• practice — not a real PYQ

  The sides of a triangle are 8 cm, 15 cm and 17 cm. A rectangle of length 12 cm has the same area as the triangle. The perimeter of the rectangle is

  • (a)40 cm
  • (b)34 cm
  • (c)17 cm
  • (d)60 cm
  Answer(2) — 8² + 15² = 289 = 17², so area = ½ × 8 × 15 = 60 sq cm; width = 60 ÷ 12 = 5 cm; perimeter = 2 × (12 + 5) = 34 cm.

  Option (1) is the triangle's perimeter; option (3) is half the rectangle's perimeter; option (4) is the area.

• practice — not a real PYQ

  The area of a triangle with sides 13 cm, 14 cm and 15 cm is

  • (a)84 sq cm
  • (b)91 sq cm
  • (c)42 sq cm
  • (d)105 sq cm
  Answer(1) — s = 21, so area = √(21 × 8 × 7 × 6) = √7056 = 84 sq cm.

  The triangle is not right-angled (13² + 14² ≠ 15²), so options (2) and (4), which take ½ × 13 × 14 and ½ × 14 × 15, fail; option (3) is half the area.