### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #271 : Plane Geometry

Give the area of the above parallelogram if .

**Possible Answers:**

**Correct answer:**

Multiply height by base to get the area.

By the 30-60-90 Theorem:

and

The area is therefore

### Example Question #1 : Parallelograms

Give the area of the above parallelogram if .

**Possible Answers:**

**Correct answer:**

Multiply height by base to get the area.

By the 45-45-90 Theorem,

.

The area is therefore

### Example Question #3 : Parallelograms

Three of the vertices of a parallelogram on the coordinate plane are . What is the area of the parallelogram?

**Possible Answers:**

Insufficient information is given to answer the problem.

**Correct answer:**

As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:

Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.

The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:

The area of the 12 by 12 square is

The area of the green triangle is .

The area of the blue triangle is .

The area of the pink triangle is .

The area of the main triangle is therefore

The parallelogram has area twice this, or .

### Example Question #4 : Parallelograms

One of the sides of a square on the coordinate plane has an endpoint at the point with coordinates ; it has the origin as its other endpoint. What is the area of this square?

**Possible Answers:**

**Correct answer:**

The length of a segment with endpoints and can be found using the distance formula with , , :

This is the length of one side of the square, so the area is the square of this, or 41.