# ISEE Upper Level Math : Triangles

## Example Questions

### Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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?

Explanation:

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!

### Example Question #61 : Isee Upper Level (Grades 9 12) Mathematics Achievement

In a right triangle, if the base of the triangle is 3, and the height is 5, what must be the length of the hypotenuse?

Explanation:

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.

### Example Question #1 : How To Find The Perimeter Of A Right Triangle

One angle of a right triangle measures 45, and the hypotenuse length is 6 centimeters. Give the perimeter of the triangle.

Explanation:

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:

### Example Question #1 : How To Find The Perimeter Of A Right Triangle

One leg of a right triangle measures ; the hypotenuse measures . What is the perimeter of the triangle?

Explanation:

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

Add the three sidelengths to get the perimeter:

### Example Question #3 : How To Find The Perimeter Of A Right Triangle

is a right triangle with altitude . If , give the perimeter of .

Explanation:

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 :

### Example Question #1 : How To Find The Height Of A Right Triangle

Note: Figure NOT drawn to scale.

In the above figure, .

Which of the following comes closest to the length of ?

Explanation:

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.

### Example Question #2 : How To Find The Height Of A Right Triangle

Note: Figure NOT drawn to scale.

In the above figure, .

Which of the following comes closest to the length of ?

Explanation:

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.

### Example Question #3 : How To Find The Height Of A Right Triangle

Refer to the above figure. Evaluate the length of  in terms of .

Explanation:

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

.

### Example Question #1 : How To Find The Height Of A Right Triangle

Refer to the above figure. Evaluate the length of  in terms of .

Explanation:

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,

### Example Question #1 : How To Find The Height Of A Right Triangle

Note: Figure NOT drawn to scale.

In the above right triangle, . Give the length of .