Chapter 4 · Geometry and Equations · Concept 1
4-1 Internal and External Division on a Number Line
A surgeon guiding a robotic arm to a tumour needs every landmark on a grid, with an exact address. Everything in this chapter starts here: giving a position a number, and then splitting the distance between two of them.
Guiding question
How does putting numbers on space let you split a path exactly?
Learning outcome. Find the coordinates of the point that divides a line segment internally or externally on a number line.
Distance first, then splitting it
A point P with coordinate a on a number line is written P(a), and the distance between A(a) and B(b) is
AB = |b − a|
The bars are there because a distance cannot be negative. Take them off and 8 − (−4) and (−4) − 8 would disagree.
internally
P is on the segment AB with AP : PB = m : n.
P(na + mbm + n)
externally
Q is on the extension of AB with AQ : QB = m : n.
Q(−na + mbm − n)
The two formulas are almost the same — which is the danger.
Internal adds n on the bottom and adds na on top. External subtracts both. Two sign changes, in the same two places. And in each of them, a goes with n and b goes with m — the letters cross over, which the book calls out in its margin.
The midpoint is not a separate formula. It is the internal one with m = n = 1, so M = (a + b)/2. Everything in this lesson is really the same rule seen from different angles.
The inside point behaves exactly as you would guess. The outside one does not — so drag the ratio and watch where it goes.
A(−4) and B(8), ratio 2 : 1
inside
(1 × −4 + 2 × 8) ÷ 3 = 4
AP : PB = 2 : 1, and P is always somewhere between the two.
outside
(−1 × −4 + 2 × 8) ÷ (2 − 1) = 20
m is bigger, so Q sits beyond B.
Try dragging m and n so they are nearly equal — 3 : 2, then 4 : 3, then 5 : 4. The external point runs further and further away, and at m = n it has gone entirely.
A(−4) and B(8). Find the coordinate of D, which divides AB externally in the ratio 2 : 3.
- 1
Name the four numbers before touching the formula.
a = −4, b = 8, m = 2, n = 3m goes with b and n goes with a — they cross over. Writing them down first stops the swap.
Work with a partner. Draw the number line before you calculate, and mark roughly where you expect each answer to land. Then check the arithmetic agrees with the picture — an external point that comes out between A and B is definitely wrong.
A(−5) and B(7).
Check (c) against the picture.
The ratio is 2 : 5, so n is the bigger one and D belongs out past A. A is at −5 and D is at −13 ✓ If your answer had landed at +13, the picture would have told you before the marker did.
1. A(−2) and B(6).
2. A(4) and B(10).
A deep-sea research habitat. Marine biologists track positions along a vertical trench wall using one axis measured in kilometres from sea level (0 km). Pod A sits at a depth of −8 km and Pod B, nearer the surface, at −3 km.
Pairing a with m instead of n.
It is na + mb — a goes with n and b goes with m. The book prints a note about this order because everyone crosses them the wrong way once.
Using the internal formula for an external point.
External has minus signs in both places: (−na + mb) over (m − n). Two changes, not one.
Losing a sign on a negative coordinate.
−3 × (−4) is +12. Keep the brackets round the coordinate all the way through.
Forgetting the bars on a distance.
AB = |b − a|. A distance is never negative, whichever way round the two points come.
Not checking against a sketch.
An internal point must land between A and B, and an external one must land outside. Half a minute of drawing catches nearly every slip.
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