G11 - Geometry problems on coordinate axes

G11 - Geometry problems on coordinate axes

Use coordinates to work out lengths along grid lines, locate midpoints and complete shapes. Check geometric claims with exact coordinate evidence, rather than relying on how a sketch looks.

Horizontal and vertical lengths

A rectangle has corners (−4,−2)(-4,-2), (3,−2)(3,-2), (3,5)(3,5) and (−4,5)(-4,5). Is it also a square? A sketch may suggest an answer, but the coordinates can settle it exactly.

You already know

In (x,y)(x,y), the horizontal coordinate comes first. Points with the same yy-coordinate lie on a horizontal line; points with the same xx-coordinate lie on a vertical line. Subtracting a negative adds its positive opposite.

On a horizontal segment, only xx changes. On a vertical segment, only yy changes. The segment length is the larger changing coordinate minus the smaller one. Crossing an axis does not change this rule.

[DIAGRAM: asset_name: Distances across the axes; asset_slug: aqa8300_g32_axis_lengths; caption: Horizontal length uses the change in x; vertical length uses the change in y. Crossing zero still means subtracting the smaller coordinate; description: An equal-scale coordinate grid shows horizontal segment from minus 3 comma 2 to 5 comma 2 and vertical segment from 1 comma minus 4 to 1 comma 6. Separate blue and sage dimension arrows show lengths 8 and 10. Endpoints are clearly labelled without covering the axes.]
Horizontal length uses the change in x

Lengths parallel to an axis

For a horizontal segment,

length=larger x−smaller x\text{length}=\text{larger }x-\text{smaller }x

For a vertical segment, use the larger yy minus the smaller yy. These rules give positive lengths for distinct endpoints.

Not on the formulae sheet: learn them.

A coordinate difference is a movement with direction. A length is a positive distance. Subtract in the larger-minus-smaller order to avoid a negative length.

Worked example 1: a horizontal side

Question. A=(−3,2)A=(-3,2) and B=(5,2)B=(5,2). Work out ABAB. Non-calculator.

Plan. The yy-coordinates agree, so use the difference in xx.

yA=yB=2horizontal segmentAB=5−(−3)=8larger x minus smaller x\begin{aligned} y_A&=y_B=2 && \text{horizontal segment}\\ AB&=5-(-3)=8 && \text{larger }x\text{ minus smaller }x \end{aligned}

Why this step? The common height 22 does not contribute to the horizontal distance.

Answer. 88 units.

Check. Moving 88 units right from x=−3x=-3 reaches x=5x=5.

The second example changes the direction to vertical. It still crosses zero; read the changing coordinate before subtracting.

Worked example 2: a vertical side crossing zero

Question. C=(1,−4)C=(1,-4) and D=(1,6)D=(1,6). Work out CDCD. Non-calculator.

Plan. The xx-coordinates agree, so subtract the smaller yy from the larger.

xC=xD=1vertical segmentCD=6−(−4)difference in height=10subtracting a negative adds\begin{aligned} x_C&=x_D=1 && \text{vertical segment}\\ CD&=6-(-4) && \text{difference in height}\\ &=10 && \text{subtracting a negative adds} \end{aligned}

Why this step? From −4-4 to 00 is 44 units, and from 00 to 66 is another 66.

Answer. 1010 units.

Check. 4+6=104+6=10.

These simple differences apply to horizontal or vertical segments. Adding both coordinate changes does not give the length of a sloping segment.

Watch out

Wrong: the distance from y=−4y=-4 to y=6y=6 is 6−4=26-4=2. Right: keep the negative sign inside brackets, 6−(−4)=106-(-4)=10.

Choose the changing coordinate for each side of the next shape.

Midpoints on a grid

Halfway along a sloping segment means halfway in each coordinate. Average the horizontal coordinates together, then average the vertical coordinates together.

Midpoint

The midpoint of a line segment is the point halfway between its endpoints. It divides the segment into two equal lengths. It need not be a grid intersection: a midpoint coordinate can be a fraction or decimal.

From (2,4)(2,4) to (10,8)(10,8), the full movement is 88 right and 44 up. Half the movement is 44 right and 22 up, reaching (6,6)(6,6). Averaging gives the same result without counting squares.

[DIAGRAM: asset_name: Half the movement in each coordinate; asset_slug: aqa8300_g32_midpoints; caption: The midpoint is halfway across and halfway up. The same rule works when coordinates are negative; description: Two equal-scale grid panels show segment from 2 comma 4 to 10 comma 8 with midpoint 6 comma 6, and segment from minus 5 comma 2 to 3 comma minus 4 with midpoint minus 1 comma minus 1. Dashed horizontal and vertical changes split into equal halves.]
The midpoint is halfway across and halfway up. The same rule works when coordinates are negative

Midpoint of a segment

For endpoints (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2), the midpoint is

(x1+x22,y1+y22)\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Not on the formulae sheet: learn it.

This works because moving half of the difference from the first coordinate gives

x1+x2−x12=x1+x22x_1+\frac{x_2-x_1}{2}=\frac{x_1+x_2}{2}

The same argument works for yy. It also explains why averaging an xx-coordinate with a yy-coordinate has no geometric meaning.

Worked example 3: two positive endpoints

Question. Work out the midpoint of A=(2,4)A=(2,4) and B=(10,8)B=(10,8). Non-calculator.

Plan. Average the two xx-coordinates and the two yy-coordinates separately.

xM=2+102=6halfway horizontallyyM=4+82=6halfway vertically\begin{aligned} x_M&=\frac{2+10}{2}=6 && \text{halfway horizontally}\\ y_M&=\frac{4+8}{2}=6 && \text{halfway vertically} \end{aligned}

Why this step? Each coordinate measures a different direction, so keep the two directions separate.

Answer. (6,6)(6,6).

Check. AA to MM and MM to BB each move 44 right and 22 up.

The next example changes the signs of some coordinates. Average signed numbers in exactly the same way.

Worked example 4: negative coordinates

Question. Work out the midpoint of C=(−5,2)C=(-5,2) and D=(3,−4)D=(3,-4). Non-calculator.

Plan. Add each pair with its signs, then divide by 22.

xM=−5+32=−22=−1average x-coordinatesyM=2+(−4)2=−22=−1average y-coordinates\begin{aligned} x_M&=\frac{-5+3}{2}=\frac{-2}{2}=-1 && \text{average }x\text{-coordinates}\\ y_M&=\frac{2+(-4)}{2}=\frac{-2}{2}=-1 && \text{average }y\text{-coordinates} \end{aligned}

Why this step? Taking positive distances first would lose the midpoint's position relative to the axes.

Answer. (−1,−1)(-1,-1).

Check. Both halves move 44 right and 33 down.

A midpoint can be the origin, but opposite signs alone do not guarantee that. The coordinates must cancel in each pair.

Watch out

Wrong: the midpoint of (2,4)(2,4) and (10,8)(10,8) is (3,9)(3,9), found by averaging within each point. Right: combine the two horizontal coordinates and the two vertical coordinates, giving (6,6)(6,6).

Try the signed-coordinate route with another pair.

Completing a shape from its properties

A missing vertex is fixed by the properties of the named shape. Read the vertices in their stated order: it tells you which sides are opposite.

For a rectangle with sides parallel to the axes, the missing corner combines the unused horizontal and vertical coordinates. For a sloping parallelogram, opposite sides have the same horizontal and vertical movements when read in matching directions.

[DIAGRAM: asset_name: Completing rectangles and parallelograms; asset_slug: aqa8300_g32_missing_vertices; caption: A rectangle shares horizontal and vertical coordinates. A parallelogram repeats the same movement along opposite sides; description: Two equal-scale grids show rectangle A minus 3 comma minus 2 B 5 comma minus 2 C 5 comma 4 D minus 3 comma 4, and sloping parallelogram A minus 2 comma minus 1 B 3 comma 1 C 5 comma 5 D 0 comma 3. In the second panel matching blue arrows show the move 2 right and 4 up from B to C and from A to D.]
A rectangle shares horizontal and vertical coordinates. A parallelogram repeats the same movement along opposite sides

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