MathematicsChapter 7: Coordinate Geometry
Midpoint Formula
The midpoint of the segment from (x1, y1) to (x2, y2) is ((x1+x2)/2, (y1+y2)/2).
Learning objectives
- Explain midpoint formula in clear mathematical language
- Apply the ideas in how to compute
- Solve problems involving what it means
Definition
The midpoint of the segment from (x1, y1) to (x2, y2) is (x1+x22, y1+y22). Each coordinate is the ordinary average of the two ends. The midpoint sits halfway in both the x-direction and the y-direction, so it is the balance point of the segment. Unlike distance, you add first, then divide by 2; you do not square.
Where / symbols
- ((x1+x2)/2, (y1+y2)/2)
- The midpoint of the segment from (x1, y1) to (x2, y2) is ((x1+x2)/2, (y1+y2)/2).
- average
- Each coordinate is the ordinary average of the two ends.
- halfway
- The midpoint sits halfway in both the x-direction and the y-direction, so it is the balance point of the segment.
- (4, 7)
- From (2, 4) to (6, 10) the midpoint is ((2+6)/2, (4+10)/2) = (4, 7).
Overview
Average the x-values; average the y-values:
How To Compute
Write (x1+x22, y1+y22) and keep the two halves as a pair. From (2, 4) to (6, 10) the midpoint is (2+62, 4+102) = (4, 7). From (0,0) to (8,-2) it is (4, -1). If one end is the origin, the midpoint is just half of the other pair. Fractions are allowed: (1,0) and (2,0) meet at (32, 0).
Examples (2,4) and (6,10) -> (4,7) · (0,0) and (8,-2) -> (4,-1) · (1,0) and (2,0) -> (3/2, 0)
What It Means
Halfway means each coordinate is the mean, not the difference. The midpoint divides the segment into two equal lengths: distance from A to M equals distance from M to B. That is a useful check. If a question gives a midpoint and one end, double the midpoint coordinates and subtract the known end to recover the other end.
Examples A to M equals M to B · M given, A known, find B · Origin to (6,4), midpoint (3,2)
What it means
The midpoint of the segment from (x1, y1) to (x2, y2) is (x1+x22, y1+y22).
Main ideas
1. HOW TO COMPUTE: Write (x1+x22, y1+y22) and keep the two halves as a pair. 2. WHAT IT MEANS: Halfway means each coordinate is the mean, not the difference.
Why it matters
Do not mix the midpoint with the distance formula. Distance uses squares and a square root. Midpoint uses sums and a divide by 2. Both need two points; they answer different questions: how far, versus where is the middle.
How to answer in an exam
Midpoint is (x1+x22, y1+y22). A common mix-up is to subtract instead of add, which is the start of a slope or a distance, not a midpoint. Check that M is halfway by comparing AM and MB.
Explanation
Do not mix the midpoint with the distance formula. Distance uses squares and a square root. Midpoint uses sums and a divide by 2. Both need two points; they answer different questions: how far, versus where is the middle.
In simple words
Midpoint is (x1+x22, y1+y22). A common mix-up is to subtract instead of add, which is the start of a slope or a distance, not a midpoint. Check that M is halfway by comparing AM and MB.
Key points
- (x1+x22, y1+y22)
- average
- halfway
- (4, 7)
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Midpoint Formula
Sticky Notes · Quick Revision
Core Idea
- The midpoint of the segment from (x1, y1) to (x2, y2) is (x1+x22, y1+y22).
Key Terms
- (x1+x22, y1+y22): The midpoint of the segment from (x1, y1) to (x2, y2) is (x1+x22, y1+y22).
- average: Each coordinate is the ordinary average of the two ends.
Main Pattern
- 1. HOW TO COMPUTE: Write (x1+x22, y1+y22) and keep the two halves as a pair.
- 2. WHAT IT MEANS: Halfway means each coordinate is the mean, not the difference.
Worked Reminder
- (2,4) and (6,10) -> (4,7)
- (0,0) and (8,-2) -> (4,-1)
- (1,0) and (2,0) -> (32, 0)
Common Mistake
- Midpoint is (x1+x22, y1+y22). A common mix-up is to subtract instead of add, which is the start of a slope or a distance, not a midpoint. Check that M is halfway by comparing AM and MB.
Exam Tip
- State the rule for midpoint formula, show each algebraic step, and check the final answer against the question.
60-second revision
- The midpoint of the segment from (x1, y1) to (x2, y2) is (x1+x22, y1+y22).
- Write (x1+x22, y1+y22) and keep the two halves as a pair.
- Halfway means each coordinate is the mean, not the difference.
- Do not mix the midpoint with the distance formula.
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