# ISEE Upper Level Math : Right Triangles

## Example Questions

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### Example Question #1 : Right Triangles

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

Explanation:

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?

Explanation:

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?

Explanation:

By the Pythagorean Theorem,

### Example Question #4 : Right Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Find the length of .

Explanation:

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 .

Explanation:

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 .

Explanation:

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 .

Insufficient information is given to answer the question.

Explanation:

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)

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(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 #9 : Right Triangles

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 #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 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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