What Is the Midpoint Formula and How to Find It with Solved Examples
Given any two points A and B, the line midpoint is point M that is located at halfway between points A and B.
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Observe that point M is equidistant from points A and B.
A line midpoint can only be found in a line segment. A line or ray cannot have a midpoint as the line is indefinite and can be extended indefinitely in both directions whereas a ray has only one end.
Let us now learn what is the midpoint of a line segment?
What is a Line Segment?
A line segment is a portion of a line that joins two different points.
It is the shortest distance between two points with a definite length that can be measured.
A line segment with two ending points XY is written as \[\overline{XY}\].
Define Midpoint of a Line Segment?
A midpoint of a line segment is the point on a segment that bisects the segment into two congruent segments.
The midpoint of a line segment is the point on a segment that is at the same distance or halfway between the two ending points.
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The Midpoint of a Line Segment Formula
Let (a1, b1) and (a2, b2) be the ending point of the line segment. The midpoint formula of a line segment joining these two points is given as:
Midpoint Formula
Example:
Suppose we have two points 9 and 5 on a number line, the midpoint of a line will be calculated as:
\[\frac{9 + 5}{2} = \frac{14}{2} = 7\]
Let us learn to find the midpoint of a line segment joined by the ending points (-3, 3) and (5, 3).
Let (-3, 3) be the first endpoint, so a1 = -3 and b1 = 3. Similarly, Let (5, 3) be the second endpoint, so a2 = 5 and b2 = 3. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.
Using the midpoint formula, we get:
\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (\frac{-3 + 5}{2}, \frac{3 + 3}{2}) = (\frac{2}{2}, \frac{6}{2} = (1, 3)\]
Midpoint Theorem
The statement of the midpoint theorem says that the line segment joining midpoints of the two sides of a triangle is parallel to the third side of a triangle and equal to the half of it. Consider the △ABC given below. Let points D and E be the midpoints of AB and AC. Suppose that you join the points D and E.
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The midpoint theorem says that the line DE will be parallel to the BC and equal to exactly half of BC.
How to Find the Midpoint of a Line Segment?
The midpoint of a line segment can be determined using these two different methods. These are:
Counting Method.
Using the midpoint of a line segment formula.
Counting Method
If the line segment is vertical or horizontal, you can find the midpoint of a line segment by dividing the length of a line segment by 2 and counting that value from either of the two ending points.
Midpoint Formula Method
The midpoint of a line segment that lies diagonally across the coordinate axis can be found using the midpoint formula.
The midpoint (x,y) of the line segments with ending point A (x1, y1) and B(x2, y2) can be found using the following midpoint formula.
\[(x, y) = (a, b) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})\]
Example:
Find the midpoint of segment AB, where coordinates of point A and B are (-3, 3) (1, 4) respectively.
Solution:
Using the midpoint formula, we get
\[(\frac{-3 + 1}{2}, \frac{-3 + 4}{2}) = (\frac{-2}{2}, \frac{1}{2}) = (-1, \frac{1}{2})\]
Hence, the midpoint of segment AB is (-1, ½).
The Midpoint of a Line Segment Example with Solutions
1. The diameter of a circle given below has two ending points (2, 3) and (-6, 5). Determine the coordinates of the centre of the circle given below.
Solution:
The centre of a circle divides the diameter into two equal parts. Hence, the coordinates of the centre are the midpoints of a circle.
Let (2, 3) be the first endpoint, so a1 = 2 and b1 = 3. Similarly, Let (-6, 5) be the second endpoint, so a2 = -6 and b2 = 5. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.
Using the midpoint formula, we get:
\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (\frac{2 + (-6)}{2}, \frac{-3 + 3}{2}) = (\frac{-4}{2}, \frac{2}{2}) = (-2, 1)\]
Hence, the coordinates of the centre of a circle are (-2, 1).
2. If (3, -2) is the midpoint of the line joining the points (1, x) and (5, 7). Find the value of x.
Solution:
Let (1, h) be the first endpoint, so a1 = 1 and b1 = h. Similarly, Let (5, 7) be the second endpoint, so a2 = 5 and b2 = 7. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.
Using the midpoint formula, we get:
\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (3, -2)\]
\[(\frac{1 + 5}{2}, \frac{h + 7}{2}) = (3, -2)\]
\[\frac{7 + h}{2} = -2 = 7 + h = -4\]
\[h= -11\]
Hence, the value of h is -11.
FAQs on Midpoint of a Line Segment in Coordinate Geometry
1. What is the midpoint of a line segment?
The midpoint of a line segment is the point that divides the segment into two equal parts. In coordinate geometry, it lies exactly halfway between the two endpoints.
- If a segment has endpoints A and B, the midpoint is equidistant from both.
- It represents the average of the x-coordinates and the average of the y-coordinates.
- The midpoint is commonly used in geometry, coordinate geometry, and graphing problems.
2. What is the formula for the midpoint of a line segment?
The midpoint formula is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). This formula finds the point halfway between two endpoints (x₁, y₁) and (x₂, y₂).
- Add the x-coordinates and divide by 2.
- Add the y-coordinates and divide by 2.
- The result gives the coordinates of the midpoint.
3. How do you find the midpoint between two points?
To find the midpoint between two points, apply the midpoint formula by averaging their coordinates.
- Step 1: Identify the points (x₁, y₁) and (x₂, y₂).
- Step 2: Calculate (x₁ + x₂)/2.
- Step 3: Calculate (y₁ + y₂)/2.
- Step 4: Write the midpoint as (x, y).
4. Can you give an example of finding a midpoint?
Yes, for points (2, 4) and (6, 8), the midpoint is (4, 6).
- x-coordinate: (2 + 6)/2 = 8/2 = 4
- y-coordinate: (4 + 8)/2 = 12/2 = 6
- Midpoint = (4, 6)
5. Why do we divide by 2 in the midpoint formula?
We divide by 2 because the midpoint is the average of the two endpoints. Averaging ensures the point lies exactly halfway between them.
- Add both x-values and divide by 2 to get the center horizontally.
- Add both y-values and divide by 2 to get the center vertically.
- This guarantees equal distance from both endpoints.
6. What is the midpoint formula in 1D (on a number line)?
On a number line, the midpoint is given by (a + b)/2. It represents the number exactly halfway between a and b.
- Add the two numbers.
- Divide the sum by 2.
- The result is the midpoint.
7. How is the midpoint related to the distance formula?
The midpoint divides a line segment into two equal parts, while the distance formula measures the length of those parts.
- Midpoint gives the center point.
- Distance formula: √[(x₂ − x₁)² + (y₂ − y₁)²].
- The distances from the midpoint to each endpoint are equal.
8. Can the midpoint have negative coordinates?
Yes, the midpoint can have negative coordinates if the endpoints include negative values. The formula works the same regardless of sign.
- Example: Between (−2, 4) and (2, 0).
- x-coordinate: (−2 + 2)/2 = 0
- y-coordinate: (4 + 0)/2 = 2
- Midpoint = (0, 2)
9. What is the difference between midpoint and bisector?
A midpoint is a point that divides a segment into two equal parts, while a bisector is a line or segment that cuts something into two equal parts.
- The midpoint is a single coordinate point.
- A segment bisector passes through the midpoint.
- An angle bisector divides an angle into two equal angles.
10. What are common mistakes when using the midpoint formula?
Common mistakes include forgetting to divide by 2 and mixing up coordinates. To correctly use the midpoint formula, follow these tips:
- Always pair x-values together and y-values together.
- Do not subtract; add first, then divide by 2.
- Write the final answer in coordinate form (x, y).
