Question

How do you find the midpoint and length of a line segment in a coordinate plane?

Answer 1

Hint: In the given question, we are required to find the midpoint and length of the line segment in the coordinate plane. In order to determine the length and midpoint of a line segment, we must know the coordinates of the end points of the segment. There are various methods of finding the midpoint and distance of a line segment.

Complete step by step solution:
coordinates of the both end points of the line segment.
Let the line segment be AB and the coordinates of the end points be $ A\left( {{x_1},{y_1}} \right) $ and $ B\left( {{x_2},{y_2}} \right) $ . Now, we first find out the midpoint of the line segment AB. Let the mid-point of the line segment AB be M and the coordinates of M be $ \left( {{m_1},{m_2}} \right) $ .

Now, we can find the midpoint of the line segment AB using the midpoint formula and coordinates of the endpoints of the line segment AB given to us.
So, according to the midpoint formula, coordinates of the midpoint of a line segment are $ \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) $ .
So, to find the coordinates of the midpoint of line segment AB, we calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.
So, the coordinates of the midpoint are: $ \left( {{m_1},{m_2}} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) $
Now, we have to find the length of the line segment AB.
So, let the length of the line segment be d.
Then, we can find the distance of a line segment given the endpoints of the segment using the distance formula as:
 $ d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $
So, the length of the line segment is $ d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $ .

Note: If the coordinates of the endpoints of the given line segment are in three dimensions, then we can apply the same distance formula and midpoint formula with slight variations.
The midpoint formula for three dimensions is $ M\left( {{m_1},{m_2},{m_3}} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2},\dfrac{{{z_1} + {z_2}}}{2}} \right) $ and the length of the line segment can be calculated as \[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} \].