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Year 10AC9M10A02

Distance, Midpoint and Line Relationships

Solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation.

Three formulas turn any pair of points into geometry you can prove things with.

Method

Distance
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Pythagoras in disguise.
Midpoint
M=(x1+x22,y1+y22)M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)
Average each coordinate.
Gradient
m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}
Parallel lines have equal gradients
m1=m2m_1 = m_2
Perpendicular gradients multiply to $-1$
m1m2=−1m_1 m_2 = -1
Flip it and change the sign.

Worked example

e.g. Find the distance, midpoint and gradient between A(1,2) and B(7,10).\text{Find the distance, midpoint and gradient between } A(1,2) \text{ and } B(7,10).
  1. Find the differences.Δx=6Δy=8\Delta x = 6 \quad \Delta y = 8
  2. Apply the distance formula.d=36+64=100=10d = \sqrt{36 + 64} = \sqrt{100} = 10
  3. Average the coordinates for the midpoint.M=(4,6)M = (4, 6)
  4. Divide the differences for the gradient.m=86=43m = \frac{8}{6} = \frac{4}{3}
  5. A perpendicular line flips and negates it.m⊥=−34m_\perp = -\frac{3}{4}
  6. Check: the product is −1-1.43×−34=−1✓\tfrac{4}{3} \times -\tfrac{3}{4} = -1 \quad\checkmark

Practice

Find the differences first. All three formulas use them.

1
Distance from (0,0) to (3,4)\text{Distance from } (0,0) \text{ to } (3,4)
Answer55
2
Midpoint of (2,5) and (8,1)\text{Midpoint of } (2,5) \text{ and } (8,1)
Answer(5,3)(5,3)
3
Gradient through (1,1) and (4,7)\text{Gradient through } (1,1) \text{ and } (4,7)
Answer22
4
Gradient perpendicular to m=25\text{Gradient perpendicular to } m = \tfrac{2}{5}
Answer−52-\frac{5}{2}
5
Are m=3 and m=−13 perpendicular?\text{Are } m = 3 \text{ and } m = -\tfrac{1}{3} \text{ perpendicular?}
AnswerYes. Their product is −1.\text{Yes. Their product is } -1.
6
Why is the distance formula Pythagoras?\text{Why is the distance formula Pythagoras?}
AnswerThe differences form the two shorter sides.\text{The differences form the two shorter sides.}
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