Question Video: Finding the Perimeter of a Triangle given Its Vertices Coordinates Using the Distance Formula Mathematics • Third Year of Preparatory School

Video Transcript

Given 𝐴 has coordinates four, five, 𝐵 has coordinates five, five, and 𝐶 has coordinates negative four, negative seven, what is the perimeter of triangle 𝐴𝐵𝐶?

We begin by recalling that we can calculate the distance between any two points with coordinates 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two using our knowledge of the Pythagorean theorem. The horizontal distance of our right triangle is equal to 𝑥 sub two minus 𝑥 sub one, and the vertical distance is equal to 𝑦 sub two minus 𝑦 sub one.

The line segment 𝐴𝐵 is the hypotenuse of our triangle. If we let this have length 𝑑 units, then applying the Pythagorean theorem we have 𝑑 squared is equal to 𝑥 sub two minus 𝑥 sub one all squared plus 𝑦 sub two minus 𝑦 sub one all squared. By square rooting both sides of this equation and using the fact that 𝑑 must be positive, we have the formula for the length of a line between two points on the two-dimensional coordinate plane. In this question, we will use this formula to help calculate the perimeter of triangle 𝐴𝐵𝐶.

We are given the coordinates of the three points. And since the perimeter is the distance around the outside of our triangle, we need to calculate the length of the line segments 𝐴𝐵, 𝐵𝐶, and 𝐶𝐴. It is worth noting at this point that 𝐴 and 𝐵 have the same 𝑦-coordinate. This means that they lie on a horizontal line. And we don’t actually need the distance formula to calculate the length of line segment 𝐴𝐵. It will simply be the difference between the 𝑥-coordinates. Five minus four is equal to one, and this will be the length of the side 𝐴𝐵 in our triangle.

To calculate the length of side 𝐵𝐶, we will firstly let point 𝐵 have coordinates 𝑥 sub one, 𝑦 sub one and point 𝐶 have coordinates 𝑥 sub two, 𝑦 sub two. Substituting our coordinates into the distance formula, we have 𝐵𝐶 is equal to the square root of negative four minus five all squared plus negative seven minus five all squared. Negative four minus five is negative nine, and squaring this gives us 81. Negative seven minus five is negative 12, and squaring this gives us 144. 𝐵𝐶 is therefore equal to the square root of 225. 225 is a square number as it is equal to 15 squared. This means that the square root of 225 is 15. Side length 𝐵𝐶 is equal to 15 length units.

We will now repeat this process for the side length 𝐶𝐴, where this time we will let point 𝐴 have coordinates 𝑥 sub one, 𝑦 sub one. 𝐶𝐴 is therefore equal to the square root of negative four minus four all squared plus negative seven minus five all squared. This is equal to the square root of 64 plus 144, which simplifies to the square root of 208. We can simplify this using our knowledge of the laws of radicals. Since 16 multiplied by 13 is 208, the square root of 208 can be written as the square root of 16 multiplied by the square root of 13. We know that the square root of 16 is equal to four. 𝐶𝐴 is therefore equal to four root 13.

We now have the lengths of all three sides of our triangle. The perimeter is the sum of these. It is equal to one plus 15 plus four root 13. This can be simplified to 16 plus four root 13. This is the perimeter of triangle 𝐴𝐵𝐶 in length units.