### All ISEE Middle Level Quantitative Resources

## Example Questions

### Example Question #40 : Plane Geometry

A triangle has base 80 inches and area 4,200 square inches. What is its height?

**Possible Answers:**

**Correct answer:**

Use the area formula for a triangle, setting :

inches

### Example Question #1 : How To Find The Area Of A Triangle

The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?

**Possible Answers:**

It is impossible to determine the area from the information given

**Correct answer:**

The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or

centimeters.

The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is

square centimeters.

### Example Question #2 : How To Find The Area Of A Triangle

The above figure depicts Square . , , and are the midpoints of , , and , respectively.

has area . What is the area of Square ?

**Possible Answers:**

**Correct answer:**

Since , , and are the midpoints of , , and , if we call the length of each side of the square, then

The area of is half the product of the lengths of its legs:

The area of the square is the square of the length of a side, which is . This is eight times the area of , so the correct choice is

### Example Question #3 : How To Find The Area Of A Triangle

Which of the following is the greater quantity?

(a) The area of the above triangle

(b) 800

**Possible Answers:**

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

which is less than 800.

### Example Question #4 : How To Find The Area Of A Triangle

The above figure gives the lengths of the three sides of the triangle in feet. Give its area in *square inches.*

**Possible Answers:**

**Correct answer:**

The area of a right triangle is half the product of the lengths of its legs, which here are feet and feet.

Multiply each length by 12 to convert to inches - the lengths become and . The area in square inches is therefore

square inches.

### Example Question #5 : How To Find The Area Of A Triangle

Figure NOT drawn to scale

Square has area 1,600. ; . Which of the following is the greater quantity?

(a) The area of

(b) The area of

**Possible Answers:**

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

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

**Correct answer:**

(b) is the greater quantity

Square has area 1,600, so the length of each side is .

Since ,

Therefore, .

has as its area ; has as its area .

Since and , it follows that

and

has greater area than .

### Example Question #6 : How To Find The Area Of A Triangle

Figure NOT drawn to scale.

In the above diagram, Square has area 400. Which is the greater quantity?

(a) The area of

(b) The area of

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

Square has area 400, so its common sidelength is the square root of 400, or 20. Therefore,

.

The area of a right triangle is half the product of the lengths of its legs.

has legs and , so its area is

.

has legs and , so its area is

.

has the greater area.

### Example Question #7 : How To Find The Area Of A Triangle

Figure NOT drawn to scale

The above diagram depicts Parallelogram . Which is the greater quantity?

(a) The area of

(b) The area of

**Possible Answers:**

(a) and (b) are equal

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

(a) is the greater quantity

**Correct answer:**

(a) and (b) are equal

Opposite sides of a parallelogram have the same measure, so

Base of and base of have the same length; also, as can be seen below, both have the same height, which is the height of the parallelogram.

Therefore, the areas of and have the same area - .

### Example Question #8 : How To Find The Area Of A Triangle

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

(a) The perimeter of the triangle

(b) 3 feet

**Possible Answers:**

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

The perimeter of the triangle - the sum of the lengths of its sides - is

inches.

3 feet are equivalent to inches, so this is the greater quantity.