Geometry

Triangles

Medium

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

$\frac{abr^2}{a^2+b^2}$

$\frac{abr^2}{2(a^2+b^2)}$

$\frac{2abr^2}{a^2+b^2}$

$\frac{4abr^2}{a^2+b^2}$

✅ Correct Option: 3

We have a triangle with vertices on a circle, where:

One side is a diameter of the circle (length = 2r)

The other two sides are in the ratio a : b

Any angle inscribed in a semicircle is always 90°. When we draw a triangle with one side as the diameter, the angle opposite to the diameter is always a right angle because the angle subtended by a diameter at any point on the circle is 90°.

This means our triangle is a right triangle with the diameter as the hypotenuse.

Since the two sides (other than the diameter) are in ratio a : b, let's set:

AC = ax and BC = bx for some positive value x

When two quantities are in ratio a : b, we can write them as ax and bx where x is a common factor.

Since we have a right triangle:

Hypotenuse = AB = 2r (diameter)

Other two sides = AC = ax and BC = bx

Using the Pythagorean theorem:

$AC^2 + BC^2 = AB^2$

$(ax)^2 + (bx)^2 = (2r)^2$

$a^2x^2 + b^2x^2 = 4r^2$

$x^2(a^2 + b^2) = 4r^2$

$x^2 = \dfrac{4r^2}{a^2 + b^2}$

For a right triangle, Area = $\tfrac{1}{2} \times \text{base} \times \text{height}$

Since AC and BC are the two perpendicular sides:

Area = $\dfrac{1}{2} \times AC \times BC$

Area = $\dfrac{1}{2} \times (ax) \times (bx)$

Area = $\dfrac{1}{2} \times abx^2$

Substituting our value of $x^2$ :

Area = $\dfrac{1}{2} \times ab \times \dfrac{4r^2}{a^2 + b^2}$

Area = $\dfrac{2abr^2}{a^2 + b^2}$

The area of the triangle is $\dfrac{2abr^2}{a^2 + b^2}$

This problem beautifully combines the inscribed angle theorem with the Pythagorean theorem. Remember that whenever we see a triangle with one side as a diameter, we automatically have a right triangle to work with.

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CAT Past Year Questions (Topic-Wise)

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