### All PSAT Math Resources

## Example Questions

### Example Question #522 : Geometry

How many diagonals are there in a regular hexagon?

**Possible Answers:**

**Correct answer:**

A diagonal is a line segment joining two non-adjacent vertices of a polygon. A regular hexagon has six sides and six vertices. One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals. Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

where .

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

How many diagonals are there in a regular hexagon?

**Possible Answers:**

18

3

9

10

6

**Correct answer:**

9

A diagonal connects two non-consecutive vertices of a polygon. A hexagon has six sides. There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon. However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals. An octagon has 20 diagonals.

### Example Question #1 : Hexagons

Regular Hexagon has a diagonal with length 1.

Give the length of diagonal .

**Possible Answers:**

**Correct answer:**

The key is to examine in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition,

.

Also, by symmetry,

,

so ,

and is a triangle whose long leg has length .

By the Theorem, , which is the hypotenuse of , has length times that of the long leg, so .

### Example Question #2 : Hexagons

Regular Hexagon has a diagonal with length 1.

Give the length of diagonal .

**Possible Answers:**

**Correct answer:**

The key is to examine in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition,

.

Also, by symmetry,

,

so ,

and is a triangle whose hypotenuse has length .

By the Theorem, the long leg of has length times that of hypotenuse , so .

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

Regular hexagon has side length of 1.

Give the length of diagonal .

**Possible Answers:**

**Correct answer:**

The key is to examine in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . To find we can subtract from . Thus resulting in:

Also, by symmetry,

,

so .

Therefore, is a triangle whose short leg has length .

The hypotenuse of this triangle measures twice the length of short leg , so .

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

Regular hexagon has side length 1.

Give the length of diagonal .

**Possible Answers:**

**Correct answer:**

The key is to examine in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . To find we subtract from . Thus resullting in

Also, by symmetry,

,

so ,

and is a triangle whose short leg has length .

The long leg of this triangle measures times the length of short leg , so .

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

Note: Figure NOT drawn to scale.

The perimeter of the above hexagon is 888. Also, . Evaluate .

**Possible Answers:**

Insufficient information is given to answer the problem.

**Correct answer:**

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

Along with , we now have a system of equations to solve for by adding:

### Example Question #261 : Plane Geometry

Note: Figure NOT drawn to scale.

The perimeter of the above figure is 132. What is ?

**Possible Answers:**

**Correct answer:**

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

### Example Question #1 : Hexagons

Note: Figure NOT drawn to scale.

The perimeter of the above figure is 600. The ratio of to is . Evaluate .

**Possible Answers:**

**Correct answer:**

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

Since the ratio of to is equivalent to - or

,

then

and we can substitute as follows:

### Example Question #1 : Hexagons

**Possible Answers:**

200

210

170

180

190