Length and midpoint

1. Overview

This topic covers calculating the length of a line segment and finding its midpoint, given the coordinates of its endpoints. These skills are essential for solving problems in coordinate geometry, and they often appear in questions involving shapes on the Cartesian plane, trigonometry, and vectors. You will need to memorise the formulas for both length and midpoint.

Key Definitions

• Line Segment: A part of a line that is bounded by two distinct end points.
• Coordinate: A set of values $(x, y)$ that show an exact position on a graph.
• Midpoint: The point on a line segment that is equidistant from both endpoints (the exact middle).
• Length (Distance): The straight-line measurement between two coordinates.
• Hypotenuse: The longest side of a right-angled triangle, which corresponds to the line segment when calculating length.

Core Content

There are no specific Core-only objectives for this sub-topic. All learning objectives for Length and Midpoint are part of the Extended curriculum.

Extended Content (Extended Curriculum Only)

A. Calculating the Length of a Line Segment

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is derived from Pythagoras’ Theorem ($a^2 + b^2 = c^2$). By treating the line segment as the hypotenuse of a right-angled triangle, we find the horizontal change ($\Delta x$) and the vertical change ($\Delta y$).

The Formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula is NOT provided on the IGCSE formula sheet. You must memorize it.

Worked example 1 — Basic length calculation

Question: Find the length of the line segment joining the points $P(2, -1)$ and $Q(6, 2)$.

1. Identify coordinates: $x_1 = 2, y_1 = -1$ and $x_2 = 6, y_2 = 2$.
2. Substitute into the formula: $d = \sqrt{(6 - 2)^2 + (2 - (-1))^2}$
3. Simplify the brackets: $d = \sqrt{(4)^2 + (3)^2}$
4. Square the terms: $d = \sqrt{16 + 9}$
5. Add the terms: $d = \sqrt{25}$
6. Take the square root: $d = 5$

Answer: The length of the line segment is $\boxed{5}$ units.

Worked example 2 — Length with surd form

Question: Calculate the exact length of the line segment connecting the points $A(-1, 3)$ and $B(2, -2)$. Give your answer in surd form.

1. Identify coordinates: $x_1 = -1, y_1 = 3$ and $x_2 = 2, y_2 = -2$.
2. Substitute into the formula: $d = \sqrt{(2 - (-1))^2 + (-2 - 3)^2}$
3. Simplify the brackets: $d = \sqrt{(3)^2 + (-5)^2}$
4. Square the terms: $d = \sqrt{9 + 25}$
5. Add the terms: $d = \sqrt{34}$

Answer: The exact length of the line segment is $\boxed{\sqrt{34}}$ units.

B. Finding the Midpoint of a Line Segment

The midpoint is essentially the "average" of the $x$-coordinates and the "average" of the $y$-coordinates.

The Formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

This formula is NOT provided on the IGCSE formula sheet. You must memorize it.

Worked example 3 — Basic midpoint calculation

Question: Find the coordinates of the midpoint $M$ of the line segment joining $A(-3, 8)$ and $B(5, 2)$.

1. Identify coordinates: $x_1 = -3, y_1 = 8$ and $x_2 = 5, y_2 = 2$.
2. Find the average of $x$: $x_m = \frac{-3 + 5}{2} = \frac{2}{2} = 1$
3. Find the average of $y$: $y_m = \frac{8 + 2}{2} = \frac{10}{2} = 5$
4. State as a coordinate: $M = (1, 5)$.

Answer: The midpoint is $\boxed{(1, 5)}$.

Worked example 4 — Finding an endpoint given the midpoint

Question: The midpoint of a line segment $PQ$ is $M(1, 3)$. If point $P$ has coordinates $(-2, 5)$, find the coordinates of point $Q$.

1. Identify knowns: $M(1, 3)$, $P(-2, 5)$. Let $Q = (x_2, y_2)$.
2. Midpoint formula for x: $\frac{-2 + x_2}{2} = 1$
3. Multiply both sides by 2: $-2 + x_2 = 2$
4. Add 2 to both sides: $x_2 = 4$
5. Midpoint formula for y: $\frac{5 + y_2}{2} = 3$
6. Multiply both sides by 2: $5 + y_2 = 6$
7. Subtract 5 from both sides: $y_2 = 1$
8. State as a coordinate: $Q = (4, 1)$

Answer: The coordinates of point $Q$ are $\boxed{(4, 1)}$.

---

Key Equations

Formula	Purpose	Variables
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Find distance/length	$(x_1, y_1)$, $(x_2, y_2)$ = coordinates
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Find the midpoint	$(x_1, y_1)$, $(x_2, y_2)$ = coordinates; Result is a coordinate

Note: These formulas are NOT provided on the IGCSE formula sheet. You must memorize them.

---

Common Mistakes to Avoid

• ❌ Wrong (Length): Calculating the squared distance but forgetting the square root: $(x_2 - x_1)^2 + (y_2 - y_1)^2 = \text{Length}$.
  • ✓ Right: Always ensure the final step is taking the square root: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• ❌ Wrong (Signs): Incorrectly handling negative signs when subtracting coordinates, e.g., $2 - (-3) = -1$.
  • ✓ Right: Subtracting a negative is the same as adding: $2 - (-3) = 2 + 3 = 5$. Be extremely careful with negative coordinates in the length formula. Double-check your signs!
• ❌ Wrong (Midpoint): Using subtraction instead of addition in the midpoint formula: $\left( \frac{x_2 - x_1}{2}, \frac{y_2 - y_1}{2} \right)$.
  • ✓ Right: The midpoint formula involves finding the average of the coordinates, which requires addition: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.
• ❌ Wrong (Order): Mixing $x$ and $y$ values in either formula, e.g., $\sqrt{(x_2 - y_1)^2 + (y_2 - x_1)^2}$ or $\left( \frac{x_1 + y_1}{2}, \frac{x_2 + y_2}{2} \right)$.
  • ✓ Right: Always pair $x$ with $x$ and $y$ with $y$. The formulas rely on maintaining the correct order.

---

Exam Tips

• Command Words: If the question says "Find the exact length," leave your answer in surd form (e.g., $\sqrt{29}$). If it says "Calculate the length," provide a decimal to 3 significant figures.
• Calculator Tip: When squaring negative numbers on a calculator, you must use brackets. Entering $-3^2$ gives $-9$, but $(-3)^2$ gives the correct $+9$. In the length formula, the squared results will always be positive.
• Typical Contexts: You may be asked to find the perimeter of a triangle by calculating the lengths of three different line segments using this formula.
• Working Backwards: A common exam question gives you the Midpoint and one endpoint, then asks for the other endpoint.
  • Method: If Endpoint $A = (2, 4)$ and Midpoint $M = (5, 7)$, look at the jumps. From $x=2$ to $x=5$ is $+3$, so the next $x$ is $5 + 3 = 8$. From $y=4$ to $y=7$ is $+3$, so the next $y$ is $7 + 3 = 10$. Endpoint $B = (8, 10)$.