### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #91 : Geometry

What is the slope of a line that passes through points and ?

**Possible Answers:**

**Correct answer:**

The equation for solving for the slope of a line is .

Thus, if and , then:

### Example Question #92 : Plane Geometry

Figure NOT drawn to scale

In the above figure, and are tangent segments. The ratio of the length of to that of is 5 to 3. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

(a) is the greater quantity

For the sake of simplicity, let us assume that the lengths of and are 5 and 3; this reasoning depends only on their ratio and not their actual length. The circumference of the circle is the sum of the lengths, which is 8, so and comprise and of the circle, respectively. Therefore,

; and

.

If two tangents are drawn to a circle, the measure of the angle they form is half the difference of the measures of the arcs they intercept, so

This is greater than .

### Example Question #1 : How To Find A Ray

Which is the greater quantity?

(a) The length of the line segment connecting and

(b) The length of the line segment connecting and

**Possible Answers:**

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**Correct answer:**

(a) and (b) are equal.

(a) The length of the line segment connecting and is

.

(b) The length of the line segment connecting and is

.

The segments have equal length.

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

Which is the greater quantity?

(a) The area of a regular hexagon with sidelength 1

(b) The area of an equilateral triangle with sidelength 2

**Possible Answers:**

(a) and (b) are equal

(b) is greater

(a) is greater

It is impossible to tell from the information given

**Correct answer:**

(a) is greater

A regular hexagon with sidelength can be seen as a composite of six equilateral triangles, each with sidelength . Since area is in direct proportion to the square of the sidelength, the area of the equilateral triangle with sidelength is equal to that of four of those triangles. This makes the hexagon greater in area, and it makes (a) the greater quantity.

### Example Question #92 : Geometry

Which is the greater quantity?

(a) The perimeter of a regular pentagon with sidelength 1 foot

(b) The perimeter of a regular hexagon with sidelength 10 inches

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

The sides of a regular polygon are congruent, so in each case, multiply the sidelength by the number of sides to get the perimeter.

(a) Since one foot equals twelve inches, inches.

(b) Multiply: inches

The two polygons have the same perimeter.

### Example Question #93 : Geometry

A hexagon has six angles with measures

Which quantity is greater?

(a)

(b) 240

**Possible Answers:**

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

**Correct answer:**

(a) and (b) are equal

The angles of a hexagon measure a total of . From the information, we know that:

The quantities are equal.

### Example Question #94 : Geometry

A hexagon has six angles with measures

Which quantity is greater?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

**Correct answer:**

(b) is greater.

The angles of a hexagon measure a total of . From the information, we know that:

This makes (b) greater.

### Example Question #95 : Geometry

The angles of Hexagon A measure

The angles of Octagon B measure

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

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

(B) is greater

(A) and (B) are equal

(A) is greater

**Correct answer:**

(B) is greater

The sum of the measures of a hexagon is . Therefore,

The sum of the measures of an octagon is . Therefore,

, so (B) is greater.

### Example Question #3 : Hexagons

The angles of Pentagon A measure

The angles of Hexagon B measure

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(B) is greater

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

(A) and (B) are equal

**Correct answer:**

(A) is greater

The sum of the measures of the angles of a pentagon is . Therefore,

The sum of the measures of a hexagon is . Therefore,

, so (A) is greater.

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

A regular hexagon has the same perimeter as the above right triangle. What is the length of one side of the hexagon?

**Possible Answers:**

The length cannot be determined from the information given.

**Correct answer:**

By the Pythagorean Theorem, the hypotenuse of the right triangle is

inches, making its perimeter

inches.

The regular hexagon, which has six sides of equal length, has the same perimeter, so each side measures

inches.