The hypotenuse of a right triangle can be calculated using the Pythagorean Theorem. This theorem states that if we know the lengths of the two other legs of the triangle, then we can calculate the hypotenuse. It is written in the following way:

In this formula the legs are noted by the variables,  and . The variable  represents the hypotenuse.

Substitute and solve for the hypotenuse.

Simplify.

Take the square root of both sides of the equation.

Step 1: Recall the Pythagorean theorem statement and formula.

Statement: For any right triangle, the sums of the squares of the shorter sides is equal to the square of the longest side.

Formula: In a right triangle , If  are the shorter sides and  is the longest side.. then,

Step 2: Plug in the values given to us in the problem....

Evaluate:

Simplify:

Simplify:

Take the square root...

Step 3: Simplify the root...

The length of the hypotenuse in most simplified form is  cm. 

In each choice, the two shortest sides of the triangle are 9 and 12, so the third side can be found by applying the Pythagorean Theorem. Set in the Pythagorean equation and solve for :

Take the square root:

.

The correct choice is

.

A triangle that has interior angles of  and  is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be  because a triangle's interior angles always sum to , allowing us to solve for the third angle like so:

Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.

We're told that the hypotenuse of our triangle has a length of . The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is . So, we need to set  equivalent to  and solve for .

As you can see, for this particular triangle, . Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the  angle will be equal to  inches; since , this side's length is . The side opposite the  angle will be equal to . Substituting in  into this expression, we find that this side has a length of .

Thus, the correct answer is .

Since  is a right angle - that is,  - and , it follows that

,

making  a 30-60-90 triangle. 

By the 30-60-90 Triangle Theorem, 

,

and

Refer to the diagram below:

The area of a right triangle is equal to half the product of the lengths of its legs, so

,

the correct response.

Since  is a right angle - that is,  - and , it follows that

,

making  a 30-60-90 triangle. 

By the 30-60-90 Triangle Theorem, 

,

and

Refer to the diagram below:

The perimeter - the sum of the sidelengths - is 

.