How to find the area of a parallelogram

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ISEE Upper Level Quantitative Reasoning › How to find the area of a parallelogram

Questions 1 - 4
1

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

Insufficient information is given to answer the problem.

Explanation

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

Axes_2

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:

Axes_1

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 .

2

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?

Explanation

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.

3

Parallelogram1

Give the area of the above parallelogram if .

Explanation

Multiply height by base to get the area.

By the 45-45-90 Theorem,

.

The area is therefore

4

Parallelogram2

Give the area of the above parallelogram if .

Explanation

Multiply height by base to get the area.

By the 30-60-90 Theorem:

and

The area is therefore

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