Question:
Two sides of a triangle are given. If the area of the triangle is maximum, then the angle between the given sides is
Two sides of a triangle are given. If the area of the triangle is maximum, then the angle between the given sides is
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For fixed two sides of a triangle, the area \(A=\frac{1}{2}ab\sin\theta\) is maximum when \(\sin\theta=1\), that is, when \(\theta=90^\circ\).
Updated On: Jun 26, 2026
- \(45^\circ\)
- \(30^\circ\)
- \(60^\circ\)
- \(90^\circ\)
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The Correct Option is D
Solution and Explanation
Step 1: Use the formula for area of a triangle.
If two sides of a triangle are \(a\) and \(b\), and the angle between them is \(\theta\), then the area is \[ A=\frac{1}{2}ab\sin\theta \]
Step 2: Identify the variable quantity.
Here, the two sides \(a\) and \(b\) are fixed.
So, \[ \frac{1}{2}ab \] is constant.
Therefore, the area depends only on \[ \sin\theta \]
Step 3: Find when the area is maximum.
The maximum value of \[ \sin\theta \] is \[ 1 \] This happens when \[ \theta=90^\circ \]
Step 4: Final conclusion.
Hence, the angle between the given sides is \[ \boxed{90^\circ} \]
If two sides of a triangle are \(a\) and \(b\), and the angle between them is \(\theta\), then the area is \[ A=\frac{1}{2}ab\sin\theta \]
Step 2: Identify the variable quantity.
Here, the two sides \(a\) and \(b\) are fixed.
So, \[ \frac{1}{2}ab \] is constant.
Therefore, the area depends only on \[ \sin\theta \]
Step 3: Find when the area is maximum.
The maximum value of \[ \sin\theta \] is \[ 1 \] This happens when \[ \theta=90^\circ \]
Step 4: Final conclusion.
Hence, the angle between the given sides is \[ \boxed{90^\circ} \]
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