Cut the segment into equal parts

The idea

The point P dividing the join of A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂ (PA : PB = m₁ : m₂) is ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂)).

A relay tower

A telephone company wants to put a relay tower at P on the line between towns A and B. The tower must be twice as far from B as it is from A.

Take A as the origin and 1 km as one unit. Then B is at (36, 15).

Where is P on the line from A to B?

• Halfway between A and B
• One third of the way from A to B
• Two thirds of the way from A to B

Cut AB into equal parts

Turn the dials m₁ and m₂. AB is cut into m₁ + m₂ equal parts, and P is m₁ parts from A. Look at the x-axis and the y-axis too: the same cuts split them in the same ratio.

The section formula

x = (m₁x₂ + m₂x₁) / (m₁+m₂) y = (m₁y₂ + m₂y₁) / (m₁+m₂) These are the coordinates of the point P that divides the join of A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁ : m₂, that is, PA : PB = m₁ : m₂.

Why? Cut AB into m₁ + m₂ equal parts. P is m₁ parts from A. The same cuts split the run from x₁ to x₂ and the rise from y₁ to y₂ in that ratio (similar triangles, Chapter 6). So x = x₁ + (m₁ parts of the run), which simplifies to the formula.

The tower: x = (1 × 36 + 2 × 0)/(1 + 2) = 12 and y = (1 × 15 + 2 × 0)/(1 + 2) = 5, so P = (12, 5). Each coordinate gets the *other* end's part: m₁ goes with B. A bigger m₁ makes PA longer, so P moves nearer B.

Notes

x = (m₁x₂ + m₂x₁) / (m₁+m₂) y = (m₁y₂ + m₂y₁) / (m₁+m₂) is the point dividing AB internally in the ratio m₁ : m₂ (PA : PB = m₁ : m₂).

Check yourself

Find the x-coordinate of the point that divides the join of (4, −3) and (8, 5) in the ratio 3 : 1.

Answer: 7

x = (3 × 8 + 1 × 4)/(3 + 1) = 28/4 = 7.

Now find the y-coordinate of the same point (the join of (4, −3) and (8, 5), ratio 3 : 1).

Answer: 3

y = (3 × 5 + 1 × (−3))/(3 + 1) = 12/4 = 3. So the point is (7, 3).

P divides AB in the ratio 1 : 3, so PA : PB = 1 : 3. Where is P?

The points that cut the join of A(2, −2) and B(−7, 4) into three equal parts are P and Q, with P nearer A. Find the x-coordinate of P.

Answer: -1

P divides AB in the ratio 1 : 2, so x = (1 × (−7) + 2 × 2)/(1 + 2) = −3/3 = −1. (Then y = (1 × 4 + 2 × (−2))/3 = 0, so P = (−1, 0).)

• Nearer B. PA is 1 part and PB is 3 parts, so P is closer to A, not B.
• Exactly in the middle. The middle would be 1 : 1. Here PB is 3 times PA.
• Nearer A: AB is cut into 4 equal parts, and P is 1 part from A — correct. Yes! PA is 1 part and PB is 3 parts, so P is a quarter of the way from A.