In a right-angled triangle, the product of two sides is equal to half of the square of the third side i.e., hypotenuse. One of the acute angle must be

Options

A. `60^@`
B. `30^@`
C. `45^@`
D. `15^@`

Answer

To solve the problem, we need to analyze the given condition in a right-angled triangle. Let's denote the sides of the triangle as follows:

- Let \( a \) and \( b \) be the two sides (the legs of the triangle).
- Let \( c \) be the hypotenuse.

According to the problem, we have the equation:

\[ ab = \frac{1}{2} c^2 \]

Now, we can use the Pythagorean theorem, which states that in a right-angled triangle:

\[ c^2 = a^2 + b^2 \]

### Step 1: Substitute the Pythagorean theorem into the equation

From the Pythagorean theorem, we can express \( c^2 \) in terms of \( a \) and \( b \):

\[ ab = \frac{1}{2} (a^2 + b^2) \]

### Step 2: Rearranging the equation

Now, we can rearrange this equation to isolate terms involving \( a \) and \( b \):

\[ 2ab = a^2 + b^2 \]

### Step 3: Rearranging further

Rearranging gives us:

\[ a^2 - 2ab + b^2 = 0 \]

### Step 4: Recognizing the quadratic form

This can be recognized as a quadratic equation in terms of \( a \):

\[ (a - b)^2 = 0 \]

### Step 5: Solving the quadratic equation

From the above equation, we can see that:

\[ a - b = 0 \implies a = b \]

### Step 6: Conclusion about the angles

If \( a = b \), then the triangle is an isosceles right triangle, which means that both acute angles are equal. In a right triangle, the two acute angles must add up to \( 90^\circ \). Therefore, each acute angle must be:

\[ \frac{90^\circ}{2} = 45^\circ \]

### Final Answer

Thus, one of the acute angles must be \( 45^\circ \).

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