### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : Right 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 #2 : Right 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 #3 : Right 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 #4 : Right 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 #5 : Right 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 #6 : Right Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

### Example Question #7 : Right 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,

### Example Question #8 : Right Triangles

is a right angle; , .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

**Correct answer:**

(a) and (b) are equal.

. 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 #9 : Right Triangles

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

**Possible Answers:**

**Correct answer:**

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:

Now we add them together:

Then, .

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

### 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?

**Possible Answers:**

**Correct answer:**

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 .

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