Question : In a right-angled triangle, the product of two sides is equal to half of the square of the third side, i.e. the hypotenuse. One of the acute angles must be:
Option 1: 60°
Option 2: 30°
Option 3: 45°
Option 4: 15°
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Correct Answer: 45°
Solution :
According to the question,
Given : $ab = \frac{c^{2}}{2}$...............................(i)
In $\triangle$ABC,
Using Pythagoras theorem
AC
2
= AB
2
+ BC
2
⇒ $c^{2} = a^{2}+b^{2}$ .................................. (ii)
Put the value of $c^{2}$ in equation (i)
$2ab = a^{2} + b^{2}$
⇒ $a^{2} + b^{2} - 2ab = 0$
⇒ $(a - b)^{2} = 0$
⇒ $a - b = 0$
⇒ $a = b$
$a = b$ means ABC is an isosceles right-angled triangle,
⇒ $\angle A$ = 45°, $\angle C$ = 45°
Hence, the correct answer is 45°.
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