C4.3; E4.3 - Midpoints and lengths

C4.3; E4.3 - Midpoints and lengths

The endpoints of a segment are (−4,1)(-4,1) and (2,9)(2,9). Where is its centre, and how long is it? C4.3 and E4.3 use coordinate averages for a midpoint and Pythagoras' theorem for a length, then work backwards when an endpoint is missing.

Midpoint coordinates

Halfway between x=−4x=-4 and x=2x=2 is x=−1x=-1: a horizontal change of 6 has been split into two changes of 3. Halfway between y=1y=1 and y=9y=9 is y=5y=5. Taking both halfway positions together gives the segment's midpoint, (−1,5)(-1,5).

Midpoint coordinates

The midpoint of endpoints (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) is

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

Average corresponding coordinates: horizontal with horizontal, vertical with vertical. Not on the supplied formula list for Papers 1–4: learn it.

This works because half of the horizontal change and half of the vertical change make half of the same straight segment. For example, x1+(x2−x1)/2=(x1+x2)/2x_1+(x_2-x_1)/2=(x_1+x_2)/2. The midpoint need not be the origin or a whole-number grid crossing.

[DIAGRAM: asset_name: A midpoint and its right-angled coordinate triangle; asset_slug: midpoint_length_triangle; caption: M is halfway along AB; the horizontal and vertical legs are 6 and 8 units.; description: An exact equal-scale Cartesian grid shows A at minus four comma one, B at two comma nine and M at minus one comma five. A right-angled dashed step passes from A to C at two comma one then to B; its horizontal leg is labelled six and vertical leg eight. The diagonal AB is highlighted and M is marked without hiding the line. Axis scales and point labels are clear.]
M is halfway along AB

Worked example 1: midpoint across the y-axis

Question. Find the midpoint of A(−4,1)A(-4,1) and B(2,9)B(2,9).

Plan. Average the two horizontal coordinates and the two vertical coordinates separately.

xM=−4+22=−22=−1yM=1+92=102=5\begin{aligned} x_M&=\frac{-4+2}{2}=\frac{-2}{2}=-1\\ y_M&=\frac{1+9}{2}=\frac{10}{2}=5 \end{aligned}

Answer. M=(−1,5)M=(-1,5).

Check. From AA to MM, the changes are (3,4)(3,4). From MM to BB, they are again (3,4)(3,4), so the two halves are equal and lie in the same direction.

Change only the second endpoint's horizontal coordinate. An odd horizontal total now gives a fractional midpoint coordinate.

Worked example 2: a midpoint between grid crossings

Question. Find the midpoint of A(−4,1)A(-4,1) and B(3,9)B(3,9).

Plan. Use the averages exactly, without rounding to a whole-number point.

xM=−4+32=−12yM=1+92=5\begin{aligned} x_M&=\frac{-4+3}{2}=-\frac12\\ y_M&=\frac{1+9}{2}=5 \end{aligned}

Answer. (−12,5)\left(-\frac12,5\right), or (−0.5,5)(-0.5,5).

Check. Both horizontal changes are 3.5 and both vertical changes are 4. The fractional horizontal position is necessary to split the 7-unit span equally.

Adding and dividing by 2 finds a position. Subtracting and dividing by 2 gives half a change, which must still be added to the starting coordinate if you use that route.

Watch out

The midpoint is not ((x2−x1)/2,(y2−y1)/2)((x_2-x_1)/2,(y_2-y_1)/2). For the first pair that would give (3,4)(3,4), the changes from AA to the midpoint, rather than the midpoint's address (−1,5)(-1,5).

Let the midpoint have non-integer coordinates when the calculation requires them.

Segment length

Join two endpoints and add a horizontal and a vertical side to make a right-angled triangle. The segment is the hypotenuse. Its perpendicular side lengths are the horizontal separation and vertical separation, so Pythagoras' theorem gives the length even if the segment slopes downwards.

Distance between two points

For points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2),

d=(x2−x1)2+(y2−y1)2.d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.

This is Pythagoras' theorem applied to the coordinate differences. A distance is non-negative. Not on the supplied formula list for Papers 1–4: learn the method.

A difference may be negative, but its square is positive or zero. Thus either subtraction order gives the same distance, provided each difference uses corresponding coordinates. For a horizontal or vertical segment, one difference is zero and the length is simply the absolute value of the other difference.

Worked example 3: an exact whole-number length

Question. Find the length from A(−4,1)A(-4,1) to B(2,9)B(2,9).

Plan. Use the horizontal and vertical separations as the two shorter sides of a right-angled triangle.

horizontal separation=2−(−4)=6vertical separation=9−1=8AB2=62+82=36+64=100AB=100=10\begin{aligned} \text{horizontal separation}&=2-(-4)=6\\ \text{vertical separation}&=9-1=8\\ AB^2&=6^2+8^2=36+64=100\\ AB&=\sqrt{100}=10 \end{aligned}

Answer. 10 units, completing the opening problem.

Check. Each half of the diagram has perpendicular changes 3 and 4, giving length 32+42=5\sqrt{3^2+4^2}=5. Two halves make 10 units, agreeing with the whole segment.

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