Distance and midpoint | Year 11

01 / LEARN

The idea, step by step

Distance comes from a right triangle

Joining two points creates horizontal and vertical changes. These changes form the legs of a right triangle, with the segment as its hypotenuse. Pythagoras gives the distance formula. Squaring removes the sign of each difference, so reversing the point order leaves the length unchanged.

Keep an exact square-root answer unless rounding is requested. The formula assumes that the axes use compatible length units. If one axis is hours and the other is dollars, a geometric distance in that mixed diagram normally has no useful physical interpretation.

d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

A midpoint averages matching coordinates

The midpoint is halfway along a segment, so average the two x-coordinates and average the two y-coordinates separately. It is a coordinate pair, not a length. A midpoint can have fractional coordinates even when both endpoints are at grid intersections.

These ideas help prove geometric claims. Equal distances can show equal sides or diagonals, while matching diagonal midpoints can show that diagonals bisect one another. A diagram suggests a claim; a coordinate calculation supplies evidence for the particular points.

M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

A common trap

Averaging an x-coordinate with a y-coordinate, or leaving a negative distance.

Instead: Match x with x and y with y for midpoint. Distance is a non-negative length obtained from a square root.

02 / WORKED EXAMPLES

See the thinking

Example 1

Length and midpoint

Find the distance and midpoint between A(-2,1) and B(4,9).

The changes are 6 and 8, giving $d = \sqrt{36 + 64} = 10$ .
The midpoint is $((- 2 + 4) / 2, (1 + 9) / 2) = (1, 5)$ .

Distance 10; midpoint (1,5).

Example 2

Recover an endpoint

A segment has endpoint P(3,-1) and midpoint M(5,2). Find the other endpoint Q.

If Q=(a,b), then (3+a)/2=5 and (-1+b)/2=2.
Solve the two equations to obtain a=7 and b=5.
Check: the displacement P to M equals M to Q, namely (2,3).

Q(7,5).

Check your understanding

Find the midpoint of (-4,7) and (2,-1).

Show answer

(-1,3).

Average each coordinate: (-4+2)/2=-1 and (7-1)/2=3.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the distance between (1,2) and (4,6).
2 marks
Find the exact distance between (-1,3) and (4,5).
2 marks
Find the midpoint of (-5,-2) and (3,8).
2 marks
A segment has endpoint (2,7) and midpoint (-1,4). Find its other endpoint.
2 marks
Show that the triangle with vertices (0,0), (4,0) and (2,3) is isosceles.
2 marks
A map uses kilometres on both axes. Two stations are at (-3,2) and (5,-4). Find their straight-line separation and explain why this may not be the road distance.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the distance between (1,2) and (4,6).
2 marks
$d = \sqrt{3^{2} + 4^{2}} = 5$ .
$5$
Find the exact distance between (-1,3) and (4,5).
2 marks
$d = \sqrt{5^{2} + 2^{2}} = \sqrt{29}$ .
$\sqrt{29}$
Find the midpoint of (-5,-2) and (3,8).
2 marks
((-5+3)/2,(-2+8)/2)=(-1,3).
(-1,3).
A segment has endpoint (2,7) and midpoint (-1,4). Find its other endpoint.
2 marks
Let Q=(a,b). Then 2+a=-2 and 7+b=8, so a=-4 and b=1.
(-4,1).
Show that the triangle with vertices (0,0), (4,0) and (2,3) is isosceles.
2 marks
Each sloping side has horizontal change of magnitude 2 and vertical change of magnitude 3.
Their equal lengths establish the isosceles claim.
The two sloping sides both have length $\sqrt{13}$ .
A map uses kilometres on both axes. Two stations are at (-3,2) and (5,-4). Find their straight-line separation and explain why this may not be the road distance.
2 marks
$d = \sqrt{8^{2} + {(- 6)}^{2}} = 10$ km.
The formula measures the direct geometric distance, not a route following roads.
10 km; roads need not follow the straight segment.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

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