Consider the triangle below.

- How can we find the distance from A to C?
- How can we find the distance from B to C?
- How can we find the distance from A to B?

We can use the **Pythagorean theorem** to help us find distances and lengths on the coordinate plane.

This equation is known as the **distance formula**.

\displaystyle d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}

\bm{d}

distance between the two points

\bm{\left(x_1,y_1\right)}

coordinates of the first point

\bm{\left(x_2,y_2\right)}

coordinates of the second point

Find the distance between A \left(-1,9\right) and B \left(-4,1\right). Leave your answer in exact form.

Worked Solution

Idea summary

The **distance formula**:

\displaystyle d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}

\bm{d}

distance between the two points

\bm{\left(x_1,y_1\right)}

coordinates of the first point

\bm{\left(x_2,y_2\right)}

coordinates of the second point

To calculate the area and perimeter of polygons on the coordinate plane, we can first use the distance formula to find the relevant distances.

We define a square with side lengths of 1 unit to have an area of 1 square unit. With this definition we can easily find that the area of a rectangle will be the product of its length and width, and we can then use this to establish area formulas for a number of other polygons:

A=lw

A=\dfrac{1}{2}bh

Consider the polygon shown below.

- How could we find the perimeter of this shape?
- How could we find the area of this shape?

For figures which are not one of our basic shapes, we can break down the **composite figure** into basic shapes, then add or subtract the pieces.

Calculate the area of the triangle shown below.

Worked Solution

Consider a quadrilateral with vertices A \left(-4,3\right), B \left(-2,-4\right), C \left(4,-4\right), and D \left(5,3\right).

a

Determine the perimeter of the quadrilateral, rounding your answer to two decimal places.

Worked Solution

b

Determine the area of the quadrilateral.

Worked Solution

Idea summary

To find the perimeter of a shape, we can use the distance formula to identify the side lengths. Then we find the sum of all of the side lengths.

To find the area of a shape, we can break the shape into familiar polygons. We can then find the areas of those polygons and add them together.