# HiSET: Math : Special triangles

## Example Questions

### Example Question #1 : Special Triangles

Two of a triangle's interior angles measure  and , respectively. If this triangle's hypotenuse is  long, what are the lengths of its other sides?

Explanation:

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 .

### Example Question #1 : Special Triangles

Examine the above triangle. Which of the following correctly gives the area of ?

None of the other choices gives the correct response.

Explanation:

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.

### Example Question #1 : Special Triangles

Examine the above triangle. Which of the following correctly gives the perimeter of ?

Explanation:

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

.