The altitude of a right triangle is $7\text{ cm}$ less than its base. If the hypotenuse is $13\text{ cm}$, the base is:
(a) $12\text{ cm}$
(b) $5\text{ cm}$
(c) $10\text{ cm}$
(d) $9\text{ cm}$
Step-by-Step Solution
Key Concept: Base $= x$, altitude $= x - 7$. $x^2 + (x - 7)^2 = 13^2 = 169 \Rightarrow 2x^2 - 14x + 49 = 169 \Rightarrow 2x^2 - 14x - 120 = 0 \Rightarrow x^2 - 7x - 60 = 0 \Rightarrow (x - 12)(x + 5) = 0 \Rightarrow x = 12$.
$x^2 - 7x - 60 = 0 \Rightarrow (x - 12)(x + 5) = 0 \Rightarrow x = 12\text{ cm}$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Solving quadratic $x = 12\text{ cm}$: 1.0 Mark
Correct Answer: $12\text{ cm}$