Show that for any p and q on the line ,the distance from p+q/2 is the same as the distance from p+q/2 to q.
To show that the distance from point p+q/2 to point p is the same as the distance from point p+q/2 to point q, we can use the distance formula.
Let's assume that p and q are points on a line. To find the distance between two points, we can use the formula:
Distance = Square root of [(x2 - x1)^2 + (y2 - y1)^2]
In this case, we have point P and point Q on the line. The coordinates of point P are (x1, y1), and the coordinates of point Q are (x2, y2).
Now, let's find the distance between p+q/2 and p:
Step 1: Find the coordinates of p+q/2:
To find the coordinates of p+q/2, we add the x-coordinates and y-coordinates separately.
If p = (x1, y1) and q = (x2, y2), then p+q/2 = (x1 + x2)/2, (y1 + y2)/2.
Step 2: Calculate the distance between p+q/2 and p:
Using the distance formula mentioned earlier, we substitute the coordinates of p+q/2 and p into the formula:
Distance = Square root of [((x1 + x2)/2 - x1)^2 + ((y1 + y2)/2 - y1)^2]
Simplifying this equation gives:
Distance = Square root of [((x1 - x2)/2)^2 + ((y1 - y2)/2)^2]
Now, let's find the distance between p+q/2 and q using the same process:
Step 3: Calculate the distance between p+q/2 and q:
Using the distance formula, substitute the coordinates of p+q/2 and q into the formula:
Distance = Square root of [((x1 + x2)/2 - x2)^2 + ((y1 + y2)/2 - y2)^2]
Simplifying this equation gives us:
Distance = Square root of [((x1 - x2)/2)^2 + ((y1 - y2)/2)^2]
Comparing the two distances, we can see that they are equal:
Distance(p+q/2, p) = Distance(p+q/2, q)
Therefore, we can conclude that for any points p and q on the line, the distance from p+q/2 to p is equal to the distance from p+q/2 to q.