Finding the Distance etween Two Points In this lesson, we will be learning how to calculate the distance between two points, say and. If we do not know the coordinates of and on the artesian Plane, we can just refer to them as (x 1 ) and (x 2 ), so that we can tell them apart. (x 2 ) (x 1 ) alculating the distance between two points is not very different to calculating the hypotenuse in one of Pythagoras rightangled triangles. To find the leg, we would need to find the difference between the two x values. (x 2 ) x 1 x 2 (x 1 ) y 1 (x 2 - x 1 ) 1
To find the leg, we would need to find the difference between the two y values. alculate the distance between point (1, 2) (x 1 ) and (7, 10) (x 2 ) y 2 (x 2 ) (y 2 - y 1 ) It makes no difference to our final calculations which point we call (x 1 ) and which point we call (x 2 ). You can assign them in either order. (x 1 ) y 1 Using Pythagoras theorem c 2 = a 2 + b 2 ; here we would have: c 2 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Finally, we would take the square root of both sides, and now we have derived a formula for calculating the distance, d, between two points: The distance between the points and is 10 units. When we have negative coordinates, we need to be more cautious with our calculations. Let s see this formula in action: (x 2 ) (7, 10) (x 1 ) (1, 2) 2
alculate the distance between point (2, -1) and (-3, 6) Example (x 2 ) (-3, 6) 1. alculate the distance between the points (-3, 7) and (5, -2) correct to two decimal places. (x 2 ) (-3, 7) (2, -1) (x 1 ) (x 1 ) (5, -2) Two minuses make a plus. ny number squared is always a positive. It makes no difference to our final calculations which point we call (x 1 ) and which point we call (x 2 ). ssign these any which way Two minuses make a plus. One decimal place is accurate enough, unless the question asks you to round off in a specific way. Watch out for these instructions. negative number squared becomes a positive. The distance between the points and is approximately 8.6 units. We are now ready for more challenging examples. Note that the question instructs us to round off to two decimal places. The distance between the points and is approximately 12.04 units. 3
2. Find the perimeter of square D in the grid below: Use the distance formula to calculate the length of side : (x 2 ) (4, 7) D (0, 5) (x 1 ) D Since we are told that the shape is a square, we know that all sides are the same length. It is also on the artesian plane, so we can label the coordinates of two points, say, and. (4, 7) (0, 5) Now to calculate the perimeter of the square, we can multiply the length of one side by 4. D The perimeter of square D is 18 units. 4
3. family lives on a secluded diary farm at coordinates (-230, -116). The closest town is located at (47, -12). How far does this family have to travel to get to this town Distances are in kilometres. Using the formula: Let: (x 2 ) = (-230, -116) and (x 1 ) = (47, -12). 4. Norman was driving to his chalet for the weekend. However, when he was located at coordinates (26, -10), his gas light turned on indicating that he could only travel another 50 km before his car would run out of gas. His Global Positioning System (GPS) gave the coordinates of the closest gas station at (-9, 2). Will he be able to make it to the gas station ll distances are in kilometres. Using the distance formula: Let: (x 2 ) = (26, -10) and (x 1 ) = (-9, 2). The family must travel approximately 296 km to the closest town. Norman is 37 km away from the gas station. He has enough gas to get him there. 5
5. circle has its centre at point O (-1, 3). The point (3, 2) is on its circumference. alculate the diameter of the circle. quick sketch will help you to visualize problem. O (-1, 3) (3, 2) alculate the radius of the circle by finding the distance between the centre O and point on the circumference. O (-1, 3) (x 2 ) (x 1 ) (3, 2) 6