One unknown ratio: k : 1

The idea

Writing the ratio m₁ : m₂ as k : 1, the dividing point is ((kx₂ + x₁)/(k + 1), (ky₂ + y₁)/(k + 1)), so an unknown ratio of division can be found as the single unknown k.

The y-axis cuts a segment

The segment joining A(5, −6) and B(−1, −4) crosses the y-axis at a point P. The y-axis cuts AB into two pieces, PA and PB.

Which piece is longer?

• PA, the piece on A's side
• PB, the piece on B's side
• They are equal

Slide k until P is on the y-axis

P divides AB so that PA : PB = k : 1. That cuts AB into k + 1 equal parts, with P k parts from A. Slide k and watch the x-coordinate of P.

The ratio k : 1

x = (kx₂ + x₁) / (k + 1) y = (ky₂ + y₁) / (k + 1) These are the coordinates of the point that divides AB internally in the ratio k : 1.

Any ratio m₁ : m₂ can be written as k : 1 with k = m₁ ÷ m₂. For example, 3 : 2 is 1.5 : 1. Then there is only one unknown, k.

To find an unknown ratio, put one coordinate of P equal to the number you know and solve for k. On the y-axis, x = 0: (−k + 5)/(k + 1) = 0, so k = 5. The ratio is 5 : 1, and y = (5 × (−4) + (−6))/(5 + 1) = −13/3, so P = (0, −13/3).

Notes

x = (kx₂ + x₁) / (k + 1) y = (ky₂ + y₁) / (k + 1) is the point dividing AB in the ratio k : 1. An unknown ratio is the single unknown k.

Check yourself

The point (−4, 6) divides the join of A(−6, 10) and B(3, −8) in the ratio k : 1. Find k.

Answer: 2/7

(3k − 6)/(k + 1) = −4 gives 3k − 6 = −4k − 4, so 7k = 2 and k = 2/7. The ratio is 2 : 7.

The x-axis divides the join of A(2, −3) and B(5, 6) in the ratio k : 1. Find k.

Answer: 1/2

y = (k × 6 + (−3))/(k + 1) = 0 gives 6k = 3, so k = 1/2. The ratio is 1 : 2, and P = (3, 0).

P(5, 6) lies on the join of A(1, 2) and B(7, 8). It divides AB in the ratio k : 1. Find k.

Answer: 2

7k + 1 = 5(k + 1), so 2k = 4 and k = 2. The ratio is 2 : 1. (Check y: (8 × 2 + 2)/3 = 6.)

The ratio 3 : 4 is written as k : 1. What is k?

• 4/3. That is 4 ÷ 3, the ratio upside down. k = 3 ÷ 4.
• 3/4 — correct. Yes! 3 : 4 = (3 ÷ 4) : 1, so k = 3/4.
• 3. k is the first number divided by the second: 3 ÷ 4, not just 3.