The answer is .

Since we know that the permieter is  inches and one side is  inches, it can be determined that the remaining two sides must combine to be  inches. The ratio of the remaining two sides is  which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by  which means . The ratio of the remaining side lengths then becomes  or . We now know the 3 side lengths are .

 is the shortest side and thus the answer.

This problem represents the definition of the side lengths of an isosceles right triangle.  By definition the sides equal , , and .  However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem .

To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of , , and  degrees. This type of triangle is a special right triangle, with the relationship between the side opposite the  degree angles serving as x, and the side opposite the  degree angle serving as .

Appyling this, if we plug  in for  we get that the side opposite the right angle (aka the diagonal) is 

If the area of the square is , we know that each side of the square is , because the area of a square is .

Then, the diagonal creates two special right triangles. Knowing that the sides = , we can find that the hypotenuse (aka diagonal) is 

In order to calculate the triangle's area, we need to find the lengths of its legs.  An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as  triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

Now we can calculate the area using the formula

Now, convert to feet.

To find the area of a triangle, multiply the base by the height, then divide by 2.  Since the short legs of an isosceles triangle are the same length, we need to know only one to know the other.  Since, a short side serves as the base of the triangle, the other short side tells us the height.

In order to calculate the triangle's perimeter, we need to find the lengths of its legs.  An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as  triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.

Now we can calculate the perimeter by doubling  and adding .

An isosceles triangle is a special triangle due to the values of its angles.  These triangles are referred to as  triangles and their side lenghts follow a specific pattern that states you can calculate the length of the hypotenuse of an isoceles triangle by multiplying the length of one of the legs by the square root of 2.

Now we can calculate the perimeter by doubling  and adding .

When a triangle has two angles equal to , it must be a isosceles right triangle.

The pattern for the sides of a is .

Since two sides are equal to , this triangle will have sides of .

Add them all together to get .

An isosceles triangle is basically two right triangles stuck together. The isosceles triangle has a base of 6, which means that from the midpoint of the base to one of the angles, the length is 3. Now, you have a right triangle with a base of 3 and a height of 4. The hypotenuse of this right triangle, which is one of the two congruent sides of the isosceles triangle, is 5 units long (according to the Pythagorean Theorem).

The total perimeter will be the length of the base (6) plus the length of the hypotenuse of each right triangle (5).

5 + 5 + 6 = 16