The section formula

A phone company wants a tower P on the straight road from A(0, 0) to B(36, 15), so that PB is twice PA. Then P divides AB in the ratio 1 : 2. Where exactly is P?

'P divides AB in the ratio m₁ : m₂' means AP : PB = m₁ : m₂. A 12 cm stick cut into 8 cm and 4 cm is divided 2 : 1.

The section formula:

P = ( (m₁x₂ + m₂x₁)/(m₁ + m₂) , (m₁y₂ + m₂y₁)/(m₁ + m₂) )

Remember it by two rules:

• x's go with x's, and y's go with y's;
• cross over: m₁ multiplies the far point B, and m₂ multiplies the near point A.

Check with the tower: x = (1 × 36 + 2 × 0)/3 = 12, y = (1 × 15 + 2 × 0)/3 = 5. So P is (12, 5).

Why it works: drop lines from A, P and B to the x-axis. The small triangles are similar (Chapter 6), so the across-distances are in the same ratio m₁ : m₂ as AP : PB.

The midpoint is the 1 : 1 case: ((x₁ + x₂)/2, (y₁ + y₂)/2). Add the two x's and halve; add the two y's and halve.

Points of trisection cut a segment into three equal parts. They divide it in the ratios 1 : 2 and 2 : 1.

Read the ratio carefully. 'AP = (3/7) AB' is not 3 : 7. If AP is 3 parts out of 7, then PB is the other 4 parts, so the ratio is 3 : 4.