The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer
Let the shortest side = $x$ m.
Then longest side (hypotenuse) = $(x + 4)$ m, and third side = $(x + 4 - 2)$ = $(x + 2)$ m.
Applying Pythagoras theorem (longest side is hypotenuse):
$$x^2 + (x+2)^2 = (x+4)^2$$
$$x^2 + x^2 + 4x + 4 = x^2 + 8x + 16$$
$$x^2 - 4x - 12 = 0$$
Factorising:
$$(x - 6)(x + 2) = 0$$
$$x = 6 \quad \text{or} \quad x = -2$$
Since length cannot be negative, $x = 6$.
The three sides are:
- Shortest side = 6 m
- Third side = $6 + 2$ = 8 m
- Hypotenuse = $6 + 4$ = 10 m
Verification: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ ✓
Area $= \dfrac{1}{2} \times 6 \times 8 = 24 \text{ m}^2$
Perimeter $= 6 + 8 + 10 = 24 \text{ m}$
Difference $= 24 - 24 = \mathbf{0}$
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Explanation
- Setting up the variable: Always let the shortest side = $x$ so the other expressions stay simple.
- Key step: The longest side must be the hypotenuse in a right triangle — apply Pythagoras accordingly.
- Reject negative root: Lengths must be positive; reject $x = -2$.
- Final part: The question asks for the difference between the numerical values of area (m²) and perimeter (m) — treat them as pure numbers and subtract. Here both equal 24, so the answer is 0. This is a common trick in board questions.
- Examiners award marks for: correct equation (1), correct factorisation/roots (2), correct sides (1), correct area/perimeter difference (1).