Definition of a circle

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Question

$A B$ is a diameter of a circle and $C$ is any point on the circumference of the circle. Then,

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Solution

In $△ A B C$ , we have
$\sin θ = \frac{B C}{A B}$ and $\cos θ = \frac{A C}{A B}$
$\Rightarrow B C = A B \sin θ$ and $A C = A B \cos θ$
Let $∆$ be the area of $△ A B C$ . Then,
$Δ = \frac{1}{2} B C \times A C = \frac{1}{2} (A B)^{2} \sin θ \cos θ = \frac{1}{4} (A B)^{2} \sin 2 θ$
Clearly, it is maximum when $\sin 2 θ = 1$ is maximum i.e. $\sin 2 θ = 1 .$ In that case, $θ = π / 4$
$∴ B C = A C = \frac{A B}{\sqrt{2}}$
Hence, the triangle is isoceles,

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Definition of a circle

Remember concepts with our Masterclasses.

AB is a diameter of a circle and C is any point on the circumference of the circle. Then,

Ready to Test Your Skills?

free study material

$A B$ is a diameter of a circle and $C$ is any point on the circumference of the circle. Then,