# ISEE Middle Level Quantitative : Geometry

## Example Questions

### 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?

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

Explanation:

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.

### Example Question #241 : Plane Geometry

Note: Figure NOT drawn to scale.

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

Explanation:

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, or  of the perimeter.

### Example Question #242 : Plane Geometry

Note: Figure NOT drawn to scale

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

Explanation:

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

The perimeter of the triangle is

.

The insect has traveled  units, which is

of the perimeter.

### Example Question #251 : Geometry

Note: Figure NOT drawn to scale

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

Explanation:

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

.

The perimeter of the triangle is

.

The insect has traveled  units, or

of the perimeter.

### Example Question #642 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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

Explanation:

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?

It is impossible to determine the area from the information given

Explanation:

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 #251 : Geometry

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

has area . What is the area of Square  ?

Explanation:

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 #1 : 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

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

(b) is the greater quantity

Explanation:

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 #651 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

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

Explanation:

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 #1 : 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

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

(b) is the greater quantity

Explanation:

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 .