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Question:

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is

Explanation:

Concept: Diameter of a circle makes an angle of 90° on the circle.

∴ (2r)² = a² + b²
⇒ 4r² = a² + b²

Now, Area of the triangle = 1/2 × base × height
= 1/2 × a × b
= $\frac{ab (a^{2} + b^{2})}{2 (a^{2} + b^{2})}$
= $\frac{ab \times 4 r^{2}}{2 (a^{2} + b^{2})}$
= $\frac{2 {abr}^{2}}{a^{2} + b^{2}}$

Hence, option (a).

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Doubts

Vivek Kumar said (2024-06-11 05:48:28)

after formula 1/2 base*height how you came to next step

Reply from Admin:

Area of the triangle = 1/2 × base × height

Here, the triangle in a right triangle. In a right triangle if you consider any of the two smaller sides as the base of the triangle then the other smaller side will become the height of the triangle.

Here, if we consider side with length 'a' as the base, then side with length 'b' will become the height of the triangle.

Hope this helps.

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