Choose the the origin as one of the endpoints. What does the Distance Formula tell you about the coordinates of the other endpoint?
(0,0) and (3,4)
Practice makes perfect
Imagine a line segment in a coordinate plane that is neither horizontal or vertical and has a length of 5 units. We are asked to determine the possible coordinates of its endpoints.
First, we are going to add the x-axis and y-axis onto the graph. Now, for simplicity, let one of the endpoints be the origin. This means that its coordinates are (0,0).
It is time for us to use what we have learned about distance on the coordinate plane. For that, let's remember the Distance Formula. It is the formula for finding the distance d between two points with coordinates (x_1,y_1) and (x_2,y_2).
d=sqrt((x_2-x_1 )^2+(y_2-y_1 )^2)
Now, we know that the distance d between the endpoints of the segment is 5 units. Also, the coordinates of one of the endpoints are ( 0, 0). Let's then substitute what we know into the formula and then transform the equation.
We found that the coordinates of the second endpoint (x_y,y_2) need to satisfy the following equation. Note that it is the only restriction we have.
x_2^2+y_2^2=25
There are infinitely many pairs of numbers (x_2,y_2) that satisfy the equation. We need to find just one of them. Let's then narrow down our area of search to natural numbers. We can check that x_2=3 and _2=4 satisfy the equation.
We found a possible pair of coordinates for the second endpoint. Let's include them in our graph.
Note that this is just a sample answer. We chose the origin as one of the endpoints for simplicity — this made the calculations easier. The final answer could be any other pair of points on the coordinate plane, as long as their distance is 5.
