30-60-90 triangle:

The leg opposite the 30-degree angle is the short leg.

The leg opposite the 60-degree angle is the long leg.

The hypotenuse is twice the short leg.

If x = short leg, then hypotenuse = 2x.

The longer leg then is x*sqrt(3).

Perimeter, P1,

= x +x*sqrt(3) +2x

= x +x(1.732) +2x

= x(4.732)

Isosceles right triangle:

The two legs are equal in length.

If y = one leg, then hypotenuse = y*sqrt(2).

Perimeter, P2,

= y +y +y*sqrt(2)

= y +y +y(1.4142)

= y(3.4142)

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"... have the same perimeter, which one will have the bigger area, and why?"

If P1 = P2, then,

x(4.732) = y(3.4142)

y = x(4.732)/(3.4142)

y = x(1.386) ...........****

Let us get the areas of both right triagles in terms of x.

Area of right triangle = (1/2)(base)(altitude) = (1/2)(leg)(other leg)

30-60-90:

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

A1 = (0.866)(x^2)

isosceles:

A2 = (1/2)(y)(y)

A2 = (0.5)(y^2)

Substitute x(1.386) for y,

A2 = (0.5)(1.386x)^2

A2 = (0.96)(x^2)

So, since (A2 = 0.96x^2) is greater than (A1 = 0.866x^2), then,

Therefore, the isosceles right triangle has a bigger area. .....answer.

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"I need to come up with the perimeters for both triangles as well,..."

Since both perimeters are equal, then if you assign any value to x, (or y), you can get the perimeters.

If x = 1,

P1 = P2 = x(4.732) = 1(4.732) = 4.732

If x = 2,

P1 = P2 = x(4.732) = 2(4.732) = 9.464

If x = 8,

P1 = P2 = x(4.732) = 8(4.732) = 37.856

Based upon the given information, there is no unique value for the perimeter.