# ISEE Upper Level Math : Right Triangles

## Example Questions

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

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 #1 : 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 #41 : Isee Upper Level (Grades 9 12) Mathematics Achievement

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 : How To Find The Length Of The Side Of A Right Triangle

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 #51 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Explanation:

By the Pythagorean Theorem,

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

is a right angle; .

Which is the greater quantity?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) 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 #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 #101 : Geometry

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