# ISEE Upper Level Math : Triangles

## Example Questions

### Example Question #1 : How To Find If Right Triangles Are Similar

is a right angle; .

Which is the greater quantity?

(a)

(b)

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) and (b) are equal.

Explanation:

. Corresponding angles of similar triangles are congruent, so since  is a right angle, so is

The hypotenuse  of  is twice as long as leg ; by the  Theorem, . Again, by similiarity,

.

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

If a right triangle has a base of   and a height of , what is the length of the hypotenuse?

Explanation:

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

We know that the base is , so we can substitute in for .  We also know that the height is , so we can substitute in for .

Next we evaluate the exponents:

Then, .

is not a perfect square, so we simply write the square root as  .

### Example Question #6 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Explanation:

To solve this problem, we are going to use the Pythagorean Theorom, which states that .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

Next, we evaluate the expoenents:

Then, .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply .

### Example Question #1 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

What is the hypotenuse of a right triangle with sides 5 and 8?

undefined

Explanation:

According to the Pythagorean Theorem, the equation for the hypotenuse of a right triangle is . Plugging in the sides, we get . Solving for , we find that the hypotenuse is :

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

In a right triangle, two sides have length . Give the length of the hypotenuse in terms of .

Explanation:

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.

### Example Question #51 : Geometry

In a right triangle, two sides have lengths 5 centimeters and 12 centimeters. Give the length of the hypotenuse.

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 , lengths of the other two sides.

### Example Question #51 : Geometry

In a rectangle, the width is 6 feet long and the length is 8 feet long. If a diagonal is drawn through the rectangle, from one corner to the other, how many feet long is that diagonal?

Explanation:

Given that a rectangle has all right angles, drawing a diagonal will create a right triangle the legs are each 6 feet and 8 feet.

We know that in a 3-4-5 right triangle, when the legs are 3 feet and 4 feet, the hypotenuse will be 5 feet.

Given that the legs of this triangle are twice as long as those in the 3-4-5 triangle, it follows that the hypotense will also be twice as long.

Thus, the diagonal in through the rectangle creates a 6-8-10 triangle. 10 is therefore the length of the diagonal.

### Example Question #10 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

In a right triangle, the legs are 7 feet long and 12 feet long. How long is the hypotenuse?

Explanation:

The pythagorean theory should be used to solve this problem.

### Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

The legs of a right triangle are equal to 4 and 5. What is the length of the hypotenuse?

Explanation:

If the legs of a right triangle are 4 and 5, to find the hypotenuse, the following equation must be used to find the hypotenuse, in which c is equal to the hypotenuse:

### Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Refer to the above diagram, which depicts a right triangle. What is the value of ?