### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #1 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 5 feet and 12 feet

(b) 8 yards

**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) is greater

The length of the hypotenuse of a triangle with legs 5 feet and 12 feet is calculated using the Pythagorean Theorem, setting :

The hypotenuse is 13 feet long. The perimeter is feet, which is equal to 10 yards.

### Example Question #2 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with hypotenuse of length 25 centimeters and one leg of length 7 centimeters

(b) One-half of a meter

**Possible Answers:**

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

**Correct answer:**

(a) is greater

The length of the second leg of the triangle can be calculated using the Pythagorean Theorem, setting :

The second leg has length 24 centimeters, so the perimeter of the triangle is

centimeters.

One-half of a meter is one-half of 100 centimeters, or 50 centimeters, so (a) is greater.

### Example Question #1 : Triangles

Which is the greater quantity?

(a) The perimeter of an equilateral triangle with sidelength 30 inches

(b) The perimeter of a square with sidelength 2 feet

**Possible Answers:**

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

**Correct answer:**

(b) is greater

Each figure has sides that are congruent, so in each case, multiply the sidelength by the number of sides.

(a) The triangle has perimeter inches

(b) 2 feet are equal to 24 inches, so the square has sidelength inches.

The square has the greater perimeter.

### Example Question #632 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

is an equilateral triangle; .

Rectangle ;

Which is the greater quantity?

(a) The perimeter of

(b) The perimeter of Rectangle

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

**Correct answer:**

It is impossible to tell from the information given

(a) The perimeter of the equilateral triangle is .

(b) ; are of unknown value, but they are equal, so we will call their common length .

Rectangle has perimeter

.

Without knowing , it cannot be determined with certainty which figure has the longer perimeter. For example:

If , then

If , then

### Example Question #1 : Triangles

is an isosceles triangle; is an equilateral triangle

Which is the greater quantity?

(a) The perimeter of

(b) The perimeter of

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

**Correct answer:**

(a) is greater

(a) As an isosceles triangle, , by definition, has two congruent sides. , so either :

in which case the perimeter of is

or

in which case the perimeter of is

(b) is an equilateral triangle, so, by definition, all of its sides are congruent; its perimeter is .

Regardless of the length of , has the greater perimeter.

### Example Question #6 : Triangles

and are right triangles, with right angles , respectively.

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

(a) is the hypotenuse of , so by the Pythagorean Theorem,

(b) is a leg of , whose hypotenuse is , so by the Pythagorean Theorem,

### Example Question #7 : Triangles

Which is the greater quantity?

(a) The perimeter of a right triangle with legs of length 3 inches and 4 inches

(b) One foot

**Possible Answers:**

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

The length of the hypotenuse of a triangle with legs 3 inches and 4 inches long is calculated using the Pythagorean Theorem, setting :

The hypotenuse is 5 inches long. The perimeter is therefore inches, which is equal to one foot.

### Example Question #3 : Triangles

is a right triangle with hypotenuse 10 inches long.

The lengths of and , in inches, can both be expressed as integers.

Which is the greater quantity?

(a)

(b) The perimeter of

**Possible Answers:**

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

**Correct answer:**

(a) and (b) are equal

By the Pythagorean Theorem,

By trial and error, it can be determined that the only two positive integers that can replace and to make this equation true are 6 and 8, in either order:

Add the three side lengths to get the perimeter inches, which is equal to 2 feet.

### Example Question #2 : Triangles

An equilateral triangle has perimeter two yards. Which is the greater quantity?

(A) The length of one side of the triangle

(B) 28 inches

**Possible Answers:**

(A) and (B) are equal

(B) is greater

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

(A) is greater

**Correct answer:**

(B) is greater

An equilateral triangle has three sides of equal length; the perimeter of this triangle is two yards, which is equal to inches. One side has length inches, which is less than 28 inches, so (B) is greater.

### Example Question #5 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above triangle. An insect walks directly from B to A, then directly from A to C. What percent of the perimeter of the triangle has he walked?

**Possible Answers:**

It is impossible to answer the question from the information given.

**Correct answer:**

By the Pythagorean Theorem, the distance from A to C, which we will call , is equal to

The perimeter of the triangle is . The insect has traveled units out of 12, which is

of the perimeter.