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Coordinates: Distance and Midpoints
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Foundational Skills
Horizontal distance – first quadrant
\[ A(1,2), B(4,2) \rightarrow 3 \]
Calculate horizontal distance where points share the same y-coordinate.
Vertical distance – first quadrant
\[ C(3,1), D(3,5) \rightarrow 4 \]
Calculate vertical distance where points share the same x-coordinate.
Distance from point to x-axis
\[ P(2,4) \rightarrow 4 \]
Recognise that vertical distance to x-axis equals |y|.
Distance from point to y-axis
\[ Q(3,2) \rightarrow 3 \]
Recognise that horizontal distance to y-axis equals |x|.
Horizontal and Vertical Distance
Horizontal distance crossing y-axis
\[ E(-2,1), F(3,1) \rightarrow 5 \]
Calculate horizontal distance when points cross the y-axis.
Vertical distance crossing x-axis
\[ G(2,-1), H(2,3) \rightarrow 4 \]
Calculate vertical distance when points cross the x-axis.
Diagonal Distance
Diagonal distance (integer) – first quadrant
\[ (0,0), (3,4) \rightarrow 5 \]
Use Pythagoras to find diagonal distance with integer result.
Diagonal distance (integer) – four quadrants
\[ (-2,1), (1,-3) \rightarrow 5 \]
Use Pythagoras across quadrants with integer result.
Diagonal distance (surd answer)
\[ (0,0), (2,3) \rightarrow \sqrt{13} \]
Use Pythagoras where answer is a surd.
Diagonal distance (simplify surd)
\[ (0,0), (2,2) \rightarrow 2\sqrt{2} \]
Calculate diagonal distance and simplify the surd.
Diagonal distance (surd) – four quadrants
\[ (-2,1), (1,-1) \rightarrow \sqrt{13} \]
Use Pythagoras across quadrants where answer is a surd.
Diagonal distance (simplify) – four quadrants
\[ (-2,0), (0,2) \rightarrow 2\sqrt{2} \]
Calculate diagonal distance across quadrants and simplify the surd.
Midpoints
Midpoint of horizontal segment
\[ A(1,3), B(5,3) \rightarrow (3,3) \]
Find midpoint where y-coordinates are equal.
Midpoint of vertical segment
\[ C(2,1), D(2,5) \rightarrow (2,3) \]
Find midpoint where x-coordinates are equal.
Midpoint of diagonal (integer result)
\[ E(1,2), F(3,4) \rightarrow (2,3) \]
Find midpoint of a diagonal segment with integer coordinates.
Midpoint of diagonal (fractional result)
\[ G(1,1), H(4,2) \rightarrow (2.5,1.5) \]
Find midpoint where coordinates are non-integer.
Midpoint in four quadrants
\[ J(-2,3), K(2,-1) \rightarrow (0,1) \]
Find midpoint when points are in different quadrants.
Midpoint in four quadrants – fractional
\[ A(2,3), B(-1,2) \rightarrow (0.5,2.5) \]
Find midpoint with non-integer coordinates across quadrants.
Reverse Problems
Finding endpoint given midpoint
\[ P(1,4), M(3,2) \rightarrow Q(5,0) \]
Work backwards from midpoint to find unknown endpoint.
Finding endpoint – four quadrants
\[ P(-2,2), M(1,1) \rightarrow Q(4,0) \]
Work backwards from midpoint across quadrants.
Finding coordinate given distance
\[ A(1,2), d=4 \rightarrow x=5 \text{ or } -3 \]
Use distance to find unknown coordinates.
Timer (Optional)
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