Chapter 4 · Geometry and Equations · Concept 1
4-5 Internal and External Division Points on the Coordinate Plane
Everything from lesson 4-1, done twice — once for the x's and once for the y's. Plus one new point that a line has nowhere to put: the place a triangle balances.
Guiding question
If you can split a line, can you split a triangle?
Learning outcome. Find the coordinates of the point dividing a line segment on the coordinate plane.
The same formulas, one coordinate at a time
For A(x₁, y₁) and B(x₂, y₂):
P, dividing AB internally in m : n
(nx₁ + mx₂m + n, ny₁ + my₂m + n)
Q, dividing AB externally in m : n
(−nx₁ + mx₂m − n, −ny₁ + my₂m − n)
M, the midpoint — the internal one with m = n = 1
(x₁ + x₂2, y₁ + y₂2)
These are not new. Each one is lesson 4-1's formula written out twice — once with the x's, once with the y's. If 4-1 is solid, the only extra work here is doing it a second time without losing concentration.
And one that IS new: the centroid.
For a triangle with corners A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
G(x₁ + x₂ + x₃3, y₁ + y₂ + y₃3)
Just the average of the three corners — and the point the triangle would balance on.
The one thing that catches people is the order of the letters. Press the flip button and watch the answer jump — “divides BA” is not the same question as “divides AB”.
P divides BA internally in 1 : 2
x: (2 × 3 + 1 × -2) ÷ 3 = 1.33 · y: (2 × 7 + 1 × -3) ÷ 3 = 3.67
The first letter of the segment name is the one that goes with n.
A(−2, −3), B(3, 7), C(5, 2). Find P, which divides BA internally in the ratio 1 : 2.
- 1
Read the segment name carefully. It says BA, not AB.
first point B(3, 7), second point A(−2, −3)The first letter is the one that pairs with n. Getting this backwards is the single commonest mistake in the lesson.
Work with a partner. Plot the three points first. Then, before calculating, each of you guesses roughly where the answer should land. A guess that turns out miles from the working usually means the letters were read in the wrong order.
A(−3, 4), B(1, −4), C(−4, 3).
Question 2 lands a long way out.
Q(11, −18) is nowhere near the triangle, and that is correct — external division points usually are. The bottom came out as 2 − 3 = −1, which flipped both signs and threw the point well past B.
1. A(2, 3), B(2, −3), C(−2, 3)
2. A(−2, 1), B(6, −3), C(1, 7)
3. A(−4, 5), B(1, −5), C(3, 0)
Question 3's centroid is the origin.
−4 + 1 + 3 = 0 and 5 − 5 + 0 = 0, so G(0, 0). Nothing has gone wrong — the three corners just happen to be arranged evenly around the origin.
A(−2, 3), B(6, 1) and C(0, y), with CA = CB. Find the centroid G of triangle ABC.
Reading BA as AB.
The first letter is the one that goes with n. "Divides BA in 1 : 2" starts from B, and starting from A gives a different point entirely.
Doing both coordinates at once.
Work out the x completely, then the y completely. The m and n never change between them — only the coordinates do.
Using the internal formula for an external point.
External has minus signs on the top AND a subtraction on the bottom. Two changes, both easy to half-do.
Dividing the centroid by 2.
A centroid has three corners, so it is divided by 3. The midpoint has two and is divided by 2.
Distrusting an answer far from the triangle.
External points usually land well outside. Check it against a sketch rather than assuming it is wrong.
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