The axis is the pendicular line at the midpoint of the two extremes that identify the segment. To find its equation, all you have to do is find the coordinates of the midpoint, the slope of the line that the extremes intercept and use the anti-reciprocal to find the perpendicular. If you want to know how to find the axis of the segment passing through two points, just follow these steps.
Steps
Method 1 of 2: Gathering Information
Step 1. Find the midpoint of the two points
To find the midpoint of two points, simply enter them into the midpoint formula: [(x1 + x2) / 2, (y1 + y2) / 2]This means that you are finding the mean with respect to each of the two coordinates of both extremes, which leads to the midpoint. Suppose we are working with (x1, y 1) by coordinates of (2, 5) and (x2, y2) with coordinates (8, 3). Here's how to find the midpoint for those two points:
- [(2 + 8) / 2, (5 + 3) / 2] =
- (10 / 2, 8 / 2) =
- (5, 4)
- The midpoint coordinates of (2, 5) and (8, 3) are (5, 4).
Step 2. Find the slope of the two points:
just connect the points in the slope formula: (y2 - y1) / (x2 - x1). The slope of a line measures the vertical variation with respect to the horizontal one. Here's how to find the slope of the line that passes through the points (2, 5) and (8, 3):
- (3 - 5) / (8 - 2) =
- -2 / 6 =
-
-1 / 3
The angle coefficient of the line is -1 / 3. To find it, you need to reduce -2 / 6 to its lowest terms, -1 / 3, since both 2 and 6 are divisible by 2
Step 3. Find the opposite sign reciprocal (anti-reciprocal) of the slope of the two points:
to find it, just take the reciprocal and change the sign. The anti-reciprocal of 1/2 is -2 / 1 or simply -2; the anti-reciprocal of -4 is 1/4.
The reciprocal and opposite of -1 / 3 is 3, because 3/1 is the reciprocal of 1/3 and the sign has been changed from negative to positive
Method 2 of 2: Calculate the Line Equation
Step 1. Write the equation for a given slope line
The formula is y = mx + b where any x and y coordinate of the line is represented by "x" and "y", the "m" is the slope and "b" represents the intercept, ie where the line intersects the y axis. Once you have written this equation, you can begin to find that of the segment axis.
Step 2. Insert the anti-reciprocal in the equation, which for points (2, 5) and (8, 3) was 3
The "m" in the equation represents the slope, so put 3 in place of the "m" in the equation y = mx + b.
- 3 -> y = mx + b
- y = 3 x + b
Step 3. Replace the coordinates of the midpoint of the segment
You already know that the midpoint of points (2, 5) and (8, 3) is (5, 4). Since the axis of the segment passes through the midpoint of the two extremes, it is possible to enter the coordinates of the midpoint in the equation of the line. Quite simply, substitute (5, 4) into the x and y respectively.
- (5, 4) -> y = 3 x + b
- 4 = 3 * 5 + b
- 4 = 15 + b
Step 4. Find the intercept
You found three of the four variables in the equation of the line. You now have enough information to solve for the remaining variable, "b", which is the intercept of this line along y. Isolate variable "b" to find its value. Just subtract 15 from both sides of the equation.
- 4 = 15 + b
- -11 = b
- b = -11
Step 5. Write the segment axis equation
To write it down, you just have to insert the slope (3) and the intercept (-11) into the equation of a line. Values must not be entered in place of x and y.
- y = mx + b
- y = 3 x - 11
- The equation of the axis of the segment of extremes (2, 5) and (8, 3) is y = 3 x - 11.
