### All ISEE Upper Level Quantitative Resources

## Example Questions

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

Refer to the above right triangle. Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

By the Pythagorean Theorem,

### Example Question #2 : How To Find The Length Of The Side Of A Right Triangle

Given with right angle ,

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

The sum of the measures of the angles of a triangle is , so:

This is a triangle, so its legs and are congruent. The quantities are equal.

### Example Question #3 : How To Find The Length Of The Side Of A Right Triangle

Give the length of one leg of an isosceles right triangle whose area is the same as the right triangle in the above diagram.

**Possible Answers:**

**Correct answer:**

The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.

The triangle in the diagram has area

square inches.

An isosceles right triangle has two legs of the same length, which we will call . The area of that triangle, which is the same as that of the one in the diagram, is therefore

inches.

### Example Question #4 : How To Find The Length Of The Side Of A Right Triangle

The perimeter of a regular octagon is 20% greater than that of the above right triangle. Which is the greater quantity?

(A) The length of one side of the octagon

(B) 3 yards

**Possible Answers:**

(A) is greater

(A) and (B) are equal

(B) is greater

It is impossible to determine which is greater from the information given

**Correct answer:**

(A) and (B) are equal

By the Pythagorean Theorem, the shorter leg has length

feet.

The perimeter of the right triangle is therefore

feet.

The octagon has perimeter 20% greater than this, or

feet.

A regular octagon has eight sides of equal length, so each side of this octagon has length

feet, which is equal to 3 yards. This makes the quantities equal.

### Example Question #5 : How To Find The Length Of The Side Of A Right Triangle

The area of a square is equal to that of the above right triangle. Which is the greater quantity?

(A) The sidelength of the square

(B) 4 yards

**Possible Answers:**

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

**Correct answer:**

(B) is greater

By the Pythagorean Theorem, the shorter leg has length

feet.

The area of a triangle is equal to half the product of its base and height; for a right triangle, the legs can serve as these. The area of the above right triangle is

square feet.

The sidelength is the square root of this; , so . Therefore each sidelength of the square is just under 11 feet. 4 yards is 12 feet, so (B) is greater.

### Example Question #6 : How To Find The Length Of The Side Of A Right Triangle

Figure NOT drawn to scale.

Refer to the above triangle. Which is the greater quantity?

(a)

(b) 108

**Possible Answers:**

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

We can compare these numbers by comparing their squares.

By the Pythagorean Theorem,

Also,

, so .

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

Consider a triangle, , in which , , and . Which is the greater quantity?

(a) 55

(b)

**Possible Answers:**

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

Suppose .

By the Converse of the Pythagorean Theorem, a triangle is right if and only if the sum of the squares of the lengths of the smallest two sides is equal to the square of the longest side. Compare the quantities and

Therefore, if

, so is right, with the right angle opposite longest side . Thus, is right and has degree measure 90.

However, has degree measure *greater* than 90, so, as a consequence of the Converse of the Pythagorean Theorem and the SAS Inequality Theorem, it holds that .