Question

22) Three vertices of a triangle on a coordinate system are (1,1). (1,4), and (5,4). What is the perimeter of the triangle?

Answer 1

Given three vertices of a triangle below

`\begin{gathered} A(1,4) \\ B(5,4) \\ C(1,1) \end{gathered}`

The diagram is shown below

To find the perimeter of the triangle, using the vertices, we will find the distance between all three points

The formula to find the distance, d, between two points is

`d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`

Using the formula for the distance between two points, the distance between B and C, i.e a is

`\begin{gathered} B(5,4)\text{ and C(1,1)} \\ a=\sqrt[]{(4-1)^2+(5-1)^2} \\ a=\sqrt[]{3^2+4^2} \\ a=\sqrt[]{9+16} \\ a=\sqrt[]{25} \\ a=5\text{ units} \end{gathered}`

Distance between B and C, a, is 5 units

Using the formula for the distance between two points, the distance between A and C i.e b is

`\begin{gathered} A(1,4)\text{ and C(1,1)} \\ b=\sqrt[]{(1-1)^2+(4-1)^2} \\ b=\sqrt[]{0^2+3^2} \\ b=\sqrt[]{9} \\ b=3\text{ units} \end{gathered}`

Distance between A and C, b, is 3 units

Using the formula for the distance between two points, the distance between A and B i.e c is

`\begin{gathered} A(1,4)\text{ and B(5,4)} \\ c=\sqrt[]{(4-4)^2+(5-1)^2} \\ c=\sqrt[]{0^2+4^2} \\ c=\sqrt[]{16} \\ c=4\text{ units} \end{gathered}`

Distance between A and B, c, is 4 units

To find the perimeter, P, of a triangle, the formula is

`\begin{gathered} P=a+b+c \\ \text{Where a, b and c are the side lengths of the triangle} \end{gathered}`

Where

`\begin{gathered} a=5\text{ units} \\ b=3\text{ units and } \\ c=4\text{ units} \end{gathered}`

Substitute the values into the formula of the perimeter, P, of a triangle

`\begin{gathered} P=5+3+4=12\text{ units} \\ P=12\text{ units} \end{gathered}`

Hence, the perimeter of the triangle, P is 12 units