If the point D is on the same line as A and B, then AB will have the same gradient as AD (or BD)

First we can find the gradient of AB, using the gradient formula $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ , taking A as $(x_{1}, y_{1})$ and B as $(x_{2}, y_{2})$

$m_{A B} = \frac{4 - 9}{- 2 - 8} = \frac{- 5}{- 10} = \frac{1}{2}$

gradient of AB [1]

Now find the gradient of AD (or BD) using the gradient formula $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ , taking A as $(x_{1}, y_{1})$ and D as $(x_{2}, y_{2})$

$m_{A D} = \frac{4 - 56}{- 2 - 100} = \frac{- 52}{- 102} = \frac{52}{102}$

gradient of AD or BD [1]

We can simplify $\frac{52}{102}$ to $\frac{26}{51}$ but not to $\frac{1}{2}$

$\frac{1}{2} \neq \frac{52}{106}$ so D does not lie on the line
or gradient of AB does not equal gradient of AD, so D does not lie on the line
correct answer plus reason [1]

The method shown above is the most efficient method. However an acceptable alternative would be to find the equation of the line through A and B, then substitute D into it to see if the equation holds true for D

Coordinate Geometry (CIE IGCSE Maths: Extended)

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