Length and midpoint
Section: Coordinate geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Finding the Length of a Line Segment
The length of a line segment joining two points can be calculated directly from their coordinates, using a method based on Pythagoras' theorem.
- For two points (x_1,y_1) and (x_2,y_2), the length is √((x_2-x_1)^2+(y_2-y_1)^2)
- This formula treats the horizontal and vertical distances between the points as the two shorter sides of a right-angled triangle, with the line segment itself as the hypotenuse
- Leave the answer as a surd (in exact form) unless the question asks for a decimal, since the distance is often irrational
Worked Example: Finding the Length Between Two Points
- Question: Find the length of the line segment joining A(1, 2) and B(7, 10).
- Step 1: Horizontal distance: 7-1=6
- Step 2: Vertical distance: 10-2=8
- Step 3: Length =√(6^2+8^2)=√(36+64)=√(100)=10
- Answer: AB = 10
Common Mistakes
MistakeForgetting to square the horizontal and vertical distances before adding them, and instead adding them directly
Fixthe distance formula uses Pythagoras' theorem - both distances must be squared first, then added, then the square root taken
MistakeLosing a negative sign when one of the coordinates is negative, before squaring
Fixsince both distances are squared, the sign does not actually affect the final answer - but subtract carefully and use brackets to avoid errors along the way
Finding the Midpoint of a Line Segment
The midpoint of a line segment is the point exactly halfway between its two endpoints. It is found by averaging the x-coordinates and averaging the y-coordinates separately.
- For two points (x_1,y_1) and (x_2,y_2), the midpoint is ((x_1+x_2/2),\ (y_1+y_2/2))
- The midpoint's x-coordinate is the average of the two x-coordinates; its y-coordinate is the average of the two y-coordinates
- The midpoint always lies exactly on the line segment joining the two points, equally far from each end
Worked Example: Finding a Midpoint
- Question: Find the midpoint of the line segment joining P(-3, 5) and Q(7, -1).
- Step 1: Midpoint x-coordinate: (-3+7/2)=2
- Step 2: Midpoint y-coordinate: (5+(-1)/2)=2
- Answer: midpoint = (2, 2)
Common Mistakes
MistakeAdding the two x-coordinates and two y-coordinates but forgetting to divide by 2 at the end
Fixa midpoint is always an average - the sum of each pair of coordinates must be halved
MistakeSubtracting the coordinates instead of adding them, confusing the midpoint method with the distance or gradient method
Fixthe midpoint formula always adds the corresponding coordinates - subtraction is used for distance and gradient instead
Finding an Unknown Coordinate from a Given Distance
Extended Only
Sometimes one coordinate of a point is unknown, but the distance to another point is given. The distance formula can then be used to form an equation and solve for the missing value.
- Substitute the known and unknown coordinates into the distance formula, along with the given length
- Square both sides to remove the square root, then rearrange to isolate the unknown
- Since squaring removes the sign, there are often two possible solutions - remember to give both
Worked Example: Solving for an Unknown Coordinate
- Question: Point M has coordinates (-2, 1) and point N has coordinates (x, 6). Given that MN = 13, find the two possible values of x.
- Step 1: MN^2=(x-(-2))^2+(6-1)^2=(x+2)^2+25
- Step 2: Set MN² = 13² = 169: (x+2)^2+25=169
- Step 3: (x+2)^2=144 x+2=12
- Step 4: x=12-2=10 or x=-12-2=-14
- Answer: x = 10 or x = -14
Common Mistakes
MistakeTaking only the positive square root when solving (x+2)²=144, missing one of the two valid solutions
Fixwhenever a squared bracket is unsquared, both the positive and negative square root must be considered
MistakeSquaring the given distance incorrectly, e.g. writing 13² as 26 instead of 169
Fixsquaring means multiplying a number by itself, not doubling it - work this out carefully as a separate step
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