Right Triangle with GREATEST AREA

[Stockmoose16]

I was doing an MGMAT challenge question, and I came across the following statement:

The right triangle with the largest area will be an isosceles right triangle (where both the base and height are of equal length).

This doesn't make sense. A 30-60-90 right triangle would have an area of (X^2 * sqrt 3)/2, which is clearly greater than an isoceles right triangle (45-45-90), which would have an area of just (x^2)/2

Can anyone explain?

[gmat009]

30-60-90 right triangle

sides will be x:(sqrt 3)x:2x

A1= 1/2 *((sqrt 3)x)*x

45-45-90

sides will be x:x:((sqrt 2)x)

A2= 1/2 *x*x

A1>A2

[Stockmoose16]

30-60-90 right triangle

sides will be x:(sqrt 3)x:2x

A1= 1/2 *((sqrt 3)x)*x

45-45-90

sides will be x((sqrt 2)x)

A2= 1/2 *x*x

A1>A2

That's the same thing I showed above.

For A1: 1/2 * (sq rt 3 * X) *X = (X^2 * sqrt 3)/ 2, which would certainly be GREATER THAN A2.

According to the MGMAT question, ISOCELES TRIANGLES have the greatest area of all right triangles. You just proved that 30-60-90 triangles have the greatest area of all right triangles.

[Stockmoose16]

I was doing an MGMAT challenge question, and I came across the following statement:

Given a hypotenuse, The right triangle with the largest area will be an isosceles right triangle (where both the base and height are of equal length).

This doesn't make sense. A 30-60-90 right triangle would have an area of (X^2 * sqrt 3)/2, which is clearly greater than an isoceles right triangle (45-45-90), which would have an area of just (x^2)/2

Can anyone explain?

[stop@800]

we are comparing in the incorrects


As we are changing base and height, we need ot keep hyp same


the 30 60 90 will be (order P B H)
1 sqrt(3) 2


the 45 90 45 will be
1 1 sqrt(2)
multiply all by sqrt(2)
so we have

sqrt(2) sqrt(2) 2



now compare the areas
1 sqrt(3) 2
it will be
sqrt(3) / 2 = 1.73/2


and for
sqrt(2) sqrt(2) 2

it will be
sqrt(2) * sqrt(2) /2
= 1


so definitely 45 45 90 is greater

Hope this helps

The modified question also contains given a hyp