The diagonal of a rectangular field is $60$ metres more than the shorter side. If the longer side is $30$ metres more than the shorter side, find the sides of the field.
Step-by-Step Solution
Key Concept: Let shorter side $= x$ m. Longer side $= x + 30$, diagonal $= x + 60$. Use Pythagoras: $x^2 + (x + 30)^2 = (x + 60)^2$.
Let shorter side $= x$ m. Longer side $= x + 30$, diagonal $= x + 60$. [0.5 Mark]
Pythagoras theorem: $x^2 + (x + 30)^2 = (x + 60)^2 \Rightarrow x^2 + (x^2 + 60x + 900) = x^2 + 120x + 3600$. [1.0 Mark]
$x^2 - 60x - 2700 = 0 \Rightarrow (x - 90)(x + 30) = 0$. [1.0 Mark]
Since length cannot be negative, $x = 90$ m (rejecting $x = -30$).
Shorter side $= 90$ m, Longer side $= 90 + 30 = 120$ m. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Setting up algebraic expressions for sides and diagonal: 0.5 Mark
Forming quadratic equation via Pythagoras: 1.0 Mark
Solving quadratic equation ($x = 90$): 1.0 Mark
Stating shorter side = 90 m and longer side = 120 m: 0.5 Mark
Correct Answer: