Question

Prove that the area of a right angled triangle with given hypotenuse is maximum, if the triangle is isoceles.

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

In the given question,

Let us say that the Hypotenuse of the triangle = H

and,

The angle made by the hypotenuse with the base of the triangle be = ∅

So,

The length of base, b = H.cos∅

The length of height, h = H.sin∅

Also,

Area of the triangle =

$=\frac{1}{2}\times base\times height = \frac{1}{2}\times (H.cos \phi) \times (H.sin \phi)\\=\frac{1}{2} \times H^{2}\times sin \phi \times cos\phi =\frac{H^{2} }{4}.sin2\phi$

∵ (2sinA.cosA = sin2A)

Now,

We know that the maximum value of the sine is 1 which is possible when ∅ = 45°.

Therefore, we can say that the right-triangle will be Isosceles as it has two angles of 45 degree.

Hence, proved.