### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : Triangles

Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?

**Possible Answers:**

**Correct answer:**

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.

3 feet is equal to inches, and 4 feet is equal to inches. We choose 36 inches, since that, but not 48 inches, is a choice.

### Example Question #2 : Triangles

The triangles are similar. Solve for .

**Possible Answers:**

**Correct answer:**

Because the triangles are similar, proportions can be used to solve for the length of the side:

Cross-multiply:

### Example Question #3 : Triangles

One of the base angles of an isosceles triangle is . Give the measure of the vertex angle.

**Possible Answers:**

**Correct answer:**

The base angles of an isosceles triangle are always equal. Therefore both base angles are .

Let the measure of the third angle. Since the sum of the angles of a triangle is , we can solve accordingly:

### Example Question #4 : Triangles

A right triangle has a hypotenuse of 10 and a side of 6. What is the missing side?

**Possible Answers:**

**Correct answer:**

To find the missing side, use the Pythagorean Theorem . Plug in (remember c is always the hypotenuse!) so that . Simplify and you get Subtract 36 from both sides so that you get Take the square root of both sides. B is 8.

### Example Question #5 : Triangles

Refer to the above diagram. Which of the following quadratic equations would yield the value of as a solution?

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

### Example Question #6 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following quadratic equations would yield the value of as a solution?

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

### Example Question #7 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Find the length of .

**Possible Answers:**

**Correct answer:**

First, find .

Since is an altitude of right to its hypotenuse,

by the Angle-Angle Postulate, so

### Example Question #8 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Find the length of .

**Possible Answers:**

**Correct answer:**

First, find .

Since is an altitude of from its right angle to its hypotenuse,

by the Angle-Angle Postulate, so

### Example Question #9 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

### Example Question #10 : Triangles

A right triangle with hypotenuse is inscribed in , a circle with radius 26. If , evaluate the length of .

**Possible Answers:**

Insufficient information is given to answer the question.

**Correct answer:**

The arcs intercepted by a right angle are both semicircles, so hypotenuse shares its endpoints with two semicircles. This makes a diameter of the circle, and .

By the Pythagorean Theorem,