Length and midpoint

Length of a line segment Extended

A line segment is the part of a line between two end points. To find its length from coordinates, draw the same right-angled triangle you use for a gradient: a horizontal side, a vertical side, and the segment itself as the hypotenuse.

The horizontal side is the change in $x$ and the vertical side is the change in $y$. Pythagoras' theorem then gives the length. For the points $(x_1,y_1)$ and $(x_2,y_2)$:

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

The method, step by step:

1. Find the change in $x$ and the change in $y$.
2. Square each one.
3. Add the squares.
4. Square root the total.

Why the signs do not matter here. A change of $-8$ and a change of $8$ give the same square, 64. A length is always positive, and squaring removes the direction. You still have to subtract correctly, though: from $x=-2$ to $x=4$ is a change of $4-(-2)=6$, not 2.

You do not need to memorise the formula if you understand the triangle. Many students sketch the triangle on a quick pair of axes, label the two short sides, and use $a^2+b^2=c^2$ directly. That is exactly what the formula says.

The length triangle Extended

From $A(1,2)$ to $B(7,10)$ the horizontal side is $7-1=6$ and the vertical side is $10-2=8$. So $A{B}^2=6^2+8^2=36+64=100$ and $AB=\sqrt{100}=10$.

Exact answers and rounded answers Extended

Most pairs of points do not give a whole-number length. What you write depends on the paper and the question.

Non-calculator (Paper 2): leave the answer as a surd, in its simplest form. Look for the largest square number that divides the number under the root: $\sqrt{80}=\sqrt{16\times 5}=4\sqrt{5}$. If the question says "give your answer in the form $k\sqrt{5}$", it is telling you which surd to aim for.

Calculator (Paper 4): give the answer correct to 3 significant figures unless told otherwise: $\sqrt{80}=8.944\dots =8.94$.

Keep the exact value for as long as you can. If the length is used again, for example in a perimeter, use the full calculator value, or the surd, and round only at the very end.

Midpoint of a line segment Extended

The midpoint of a line segment is the point exactly halfway along it. Its $x$-coordinate is halfway between the two $x$-coordinates, and its $y$-coordinate is halfway between the two $y$-coordinates. Halfway between two numbers is their mean, so:

$$\text{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

Why it works. Go halfway along the segment and draw the small triangle back to the start. It is the big length triangle scaled by $\frac{1}{2}$, so it covers half the change in $x$ and half the change in $y$. Starting at $x_1$ and adding half the change, $x_1+\frac{x_2-x_1}{2}$, simplifies to $\frac{x_1+x_2}{2}$: the mean.

Note the plus signs. The midpoint adds the coordinates and halves; the length and the gradient subtract. Mixing the two up is the most common error in this topic.

Working backwards. If you know one end, $A$, and the midpoint, $M$, the other end is as far beyond $M$ as $M$ is beyond $A$. Find the step from $A$ to $M$ and take the same step again from $M$. Equivalently, $B=(2\times M)-A$, coordinate by coordinate.

Using length and midpoint together Extended

Exam questions often combine the two skills with a fact about a shape.

• Circles. If $AB$ is a diameter, the centre is the midpoint of $AB$ and the radius is half the length of $AB$. A point lies on the circle exactly when its distance from the centre equals the radius.
• Isosceles triangles. Calculate all three side lengths. Two equal lengths show the triangle is isosceles. The line from the top vertex to the midpoint of the base is then the height, which gives the area.
• Parallelograms. The diagonals of a parallelogram cut each other in half, so both diagonals have the same midpoint. That lets you find a missing vertex.
• Unknown coordinates. If a length is given and one coordinate is a letter, form an equation with Pythagoras and solve it. A square root has two signs, so there are often two answers.

In a "show that" question, give every length in full, as a surd or with the calculation shown, and finish with a sentence that states what you have shown.