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A right triangle has sides 12 cm, 16 cm and 20 cm, with the right angle between the 12 cm and 16 cm sides. A student writes . What has gone wrong?

Class 7The Triangle and its PropertiesTriangles
  • AHe has used 16 as the hypotenuse; the hypotenuse is 20, the side opposite the right angle.
  • BNothing — the equation is correct.
  • CHe should have written 20 + 12 = 16 without squaring.
  • DPythagoras cannot be used here, because 12, 16, 20 is not a Pythagorean triple.

Step-by-Step Solution

The hypotenuse is the side facing the right angle, which is the longest side, 20 cm. The correct statement is , that is 144 + 256 = 400. The student put the hypotenuse on the wrong side of the equation. Trap: slotting the sides into the formula in the order they are listed instead of identifying the hypotenuse first. (12, 16, 20 IS a triple — it is 3, 4, 5 multiplied by 4.)
Answer: A) He has used 16 as the hypotenuse; the hypotenuse is 20, the side opposite the right angle.

Key idea: Triangle inequality

**The sum of any two sides of a triangle is greater than the third side.**

So a triangle with sides 3, 4, 5 works (3+4>5, 3+5>4, 4+5>3). But sides 1, 2, 5 do NOT form a triangle — 1 + 2 = 3 is less than 5, so the two short sides can't reach across.

Use this to test whether three given lengths can be a triangle, or to bound a missing side: if two sides are 5 and 8, the third must be greater than 8 − 5 = 3 AND less than 8 + 5 = 13.

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