Q. 125.0( 3 Votes )

# Show that the rig

Answer :

Given,

• A right angle triangle is inscribed inside the circle.

• The radius of the circle is given.

Let us consider,

• ‘r’ is the radius of the circle.

• ‘x’ and ‘y’ be the base and height of the right angle triangle.

• The hypotenuse of the ΔABC = AB^{2} = AC^{2} + BC^{2}

AB = 2r, AC = y and BC = x

Hence,

4r^{2} = x^{2} + y^{2}

y^{2} = 4r^{2} – x^{2}

--- (1)

Now, Area of the ΔABC is

Now substituting (1) in the area of the triangle,

[Squaring both sides]

------ (2)

For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. __This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.__

Differentiating the equation (2) with respect to x:

[Since and if u and v are two functions of x, then ]

------ (3)

To find the critical point, we need to equate equation (3) to zero.

[as the base of the triangle cannot be negative.]

__Now to check if this critical point will determine the maximum area of the triangle, we need to check with second differential which needs to be negative.__

Consider differentiating the equation (3) with x:

----- (4)

[Since ]

Now, consider the value of

As , so the function A is maximum at .

Now substituting in equation (1):

As , the base and height of the triangle are equal, which means that two sides of a right angled triangle are equal,

Hence the given triangle, which is inscribed in a circle, is an isosceles triangle with sides AC and BC equal.

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