The hypotenuse of a 30-60-90 triangle measures twice its shorter leg; its longer leg measures  times its shorter leg. Given one side length alone, there is no indication which of the three sides it measures; but given two, one of which is twice the other, as is the case here (10 and 20), 10 must be the shorter leg and 20 the hypotenuse. The longer leg is therefore , and the perimeter is

.

Therefore, both statements together are sufficent but neither alone is sufficient.

We are given two sides of the right triangle, so in order to calculate the perimeter we must first find the length of the third side, the hypotenuse, using the Pythagorean theorem:

Now that we know the length of the third side, we can add the lengths of the three sides to calculate the perimeter of the right triangle:

As you see that a right triangle as sides 3 and 4, you should always remember that, this right triangle is a Pythagorean Triple, in other words, its sides will be in the ratio  where  is a constant, this will save you a lot of time. Here we can say that the hypotenuse will be 5. Therefore, the circumference will be . To get the final answer, we should just divide the circumference by 2 and add the perimeter, being  or 12.

Since  are right triangles, we know that  is an isosceles right triangle. So we know that the lengths of  and  are 2 cm, so we can get the length of  by using the Pythagorean Theorem:

 is the midpoint of , so the length of  is .

Now we can use the Pythagorean Theorem again to solve for : .

Finally, we have all the elements needed to solve for the area of :

Becaue this is a right triangle and since the hypotenuse is 10 and one side is 6, the other side will be 8.  This can be found using the Pythagorean Theorem, or multiplying a 3-4-5 triangle by 2 to get a 6-8-10 triangle.  Since the area of a right triangle is half of the product of the two sides, we have 

Let  be the sidelength of the square. Then its area is .

If the hypotenuse of a  triangle is , its shorter leg is half that, or ; its longer leg is  times the shorter leg, or . The area of the triangle is half the product of the legs, or 

The ratio of the area of the square to that of the triangle is

 or 

 or 

This is a right triangle with legs along the - and -axes, so the area of each can be calculated by taking one-half the product of the two legs. 

The vertical leg has length ; the horizontal leg has length .

Now calculate the area:

The equation for the area of a right triangle is:

In this case, our values are: 

Plugging this into the equation leaves us with:

which can be rewritten as 

In order to calculate the area of a right triangle, we need to know the height and the base. We are only given the length of the base, so we first need to use the Pythagorean theorem with the given hypotenuse to find the length of the height:

Now that we have the height and the base, we can plug these values into the formula for the area of a right triangle:

To find the area, use the following formula: