At your local university, there is a triangular walking path around a grassy area. If the two legs of the triangle form a ninety degree angle, and are each 30 meters long, what is the length of the hypotenuse of the triangle?

This problem asks us to find the hypotenuse of a right triangle. To do so, we can use Pythagorean Theorem.

Where a and b are the short sides of the triangle, and c is our hypotenuse.

We know a and b are both 30 meters, so plug that in and begin!

Write the formula of the Pythagorean Theorem.

The value of  is what we will be solving for.

Substitute the side lengths.

Square root both sides.

The answer is:  

This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and side length.

We can now find the perimeter since we have all three sides:

The length of the second leg can be calculated using the Pythagorean Theorem. Set :

Add the three sidelengths to get the perimeter:

An altitude of a right triangle, which here is , divides the triangle into two triangles similar to each other and to the large triangle. Therefore, their corresponding angles are equal.

 is right - and has degree measure 90 - and  has degree measure 30, so  has degree measure 60, making the triangle a 30-60-90 triangle. Therefore, the smaller triangles are as well. 

 is the short leg of , so the hypotenuse, , has twice the length of that leg, or . 

  is the long leg of ; short leg  has as its length the length of  divided by , or .

The hypotenuse of , , has as its length twice the length of short leg , which is .

Add the lengths of the sides of :

 and , so by the Pythagorean Theorem,

Because  is the altitude from the vertex  of , 

.

Therefore,

Also,

For similar reasons, 

.

Therefore, 

Of the choices given, 60 comes closest.

 and , so by the Pythagorean Theorem,

Because  is the altitude from the vertex  of , 

.

Therefore,

Also,

For similar reasons, 

.

Therefore, 

.

Of the choices given, 75 comes closest.

The altitude of a right triangle to its hypotenuse divides the triangle into two smaller trangles similar to each other and to the large triangle. 

Therefore, 

and, consequently, 

,

or, equivalently,

 by the Pythagorean Theorem, so

.

The height of a right triangle from the vertex of its right angle is the geometric mean - in this case, the square root of the product - of the lengths of the two segments of the hypotenuse that it forms. Therefore, 

The two triangles formed by an altitude from the vertex of a right triangle are similar to each other and the large triangle, so all three are 30-60-90 triangles. Take advantage of this, applying it twice.

Looking at . By the 30-60-90 Theorem, the shorter leg of a hypotentuse measures half that of the hypotenuse. . 

Now, look at . By the same theorem, 

and