Point of Division

In this lesson, we learn to find the coordinates of a point which divides a line segment into a given ratio, either internally or externally. By internally, we mean that the point of division lies between the endpoints of the line segment. A point of external division lies on an extension of the line segment in either direction, depending on the question.

Most textbooks derive a formula for the coordinates of the point we're seeking. However, this lesson will teach the concept and technique for finding the point rather than give you another formula to memorize. The formulas are there at the end of the lesson, but we're much better off working the question rather than just plugging into a template formula.

Note: If you have not yet covered the topic of similar figures (triangles in particular) you should do that before covering this lesson.

Sometimes, instead of a ratio, we get a fraction description of the line split. We must realize that if we're dividing a line segment so that point P is say 3/5 the distance from the initial point, then, one segment will be 3 parts and the other will be 2 parts of the total length. So the ratio of division is 3 : 2.

If the fraction is , the ratio is

Internal Point of Division

All the triangles in the diagram are similar.

This means that the ratio of their side lengths is constant.

Since AP = 2/5 AB, then AC = 2/5 AE

Since AP = 2/5 AB, then CP = 2/5 EB

We can find the lengths of AE and EB very easily for they're both // to axes.

So, AE = 6 – 1, (x ₂ – x ₁ ) = 5 and EB = 7 – (– 3 ), (y ₂ – y ₁ ) = 10

‹ AC = 2/5(5) = 2 and CP = 2/5(10) = 4

If we move point A 2 units to the right and 4 units up, we will be at P.

So, the coordinates of P are (1 + 2, – 3 + 4 ) = ( 3 , 1 )

What did we do?

First, we turned the proportion statement into a fraction to find what part of AB each segment is. So, the ratio of 2 : 3 became 2/5 and 3/5.

Had the ratio been a : b, our fractions would be .

Then, we found the lengths of the sides of triangle ABE, we found their lengths and added these values to the coordinates of A. This gave us the coordinates of P.

The formula says this:

If A (x ₁ , y ₁ ), and B(x ₂ , y ₂ ) are the endpoints of line segment AB and we want P to divide AB in the ratio of a : b, then the coordinates of P are:

Well isn't this exactly what we did?

Let's look at our final statement.

the coordinates of P are (1 + 2, – 3 + 4 ) = ( 3 , 1 ).

The 2 that I added to 1 (x ₁) came from of (5) -- the horizontal displacement.

The 5 came from x ₂ – x ₁ or the difference in the x coordinates.

So didn't we just do ?

a = 2, b = 3 (from ratio of 2 : 3 ),

so, , and x ₂ – x ₁ = 5

We then took of 5 and added it to the original x coordinate.

We did exactly what the formula says to do, without having to memorize the ugly thing.

The tricky part is recognizing that a ratio of 2 : 3 means that the whole is 5 parts in total and each segment must be the correct fraction of 5. In our example, that's .

note: these questions have order. We're moving from A to B because the question named the line segment AB instead of BA. Had it been BA, we'd have worked with the coordinates of B in our last step.

External Point of Division

To find the external point of division, do exactly the same as before, only now, point P is located beyond A or B depending on the order specified in the question. So the line segment AB becomes one part of the whole rather than the whole to be divided by the point of division.

Here, we know that AC = 2 units, is 1/3 of AE, so AE must be 6 units long.

Add 6 to the initial x-value to get 7 -- the x-coordinate of P.

We know that CB = 4 units, is 1/3 of EP, so EP must be 12 units long.

Add 12 to the initial y-value to get 9 -- the y-coordinate of P.

So P is at (7, 9) and the ratio of AP : BP is 3 : 2.

If you wanted to do this one by a formula, what would the formula be?

Examples

question: Find the coordinates of point P which is 1/3 the distance from A(–7, – 2) to B(2, 10).

(make yourself a diagram on graph paper while following this example -- to see the image!)

Which means we must move point A one third (1/3) the total length of the horizontal and vertical dispacements between A and B.

The horizontal distance between A and B = 2 – (–7) = 9

The vertical distance between A and B = 10 – (–2) = 12

So, the x-coordinate must move = 3 units

The y-coordinate must move = 4 units

Adding these moves to the coordinates of A gives us P at (– 4, 2).

We moved the coordinates of point A

3 units right and 4 units up respectively.

question: A landscape artist is planning to use a hedge to fence in his property. He draws a sketch of the hedge to help him decide where to put the gate. He concludes that the gate's center should be located ths of the way from the right end of the hedge as shown.

What coordinates will he assign to the center of the gate?

solution:

Notice that we put one end of the hedge at the origin (0, 0). This is a common and clever technique which enables easy calculation of horizontal and vertical displacement. Engineers, architects and map makers use it all the time. We'll use it again when we do synthetic proofs of geometry theorems in the next lesson.

Practice

1) In what ratio does point P divide segment AB? (make a diagram)

a) P(2, 0), A(–2, – 6), B(6, 6)	b) P(0, 0), A(8, -6), B(–4, 3)
c) P( , 5), A(2, 7), B(–1, – 5)	d) P(–7, – 4), A(28, 6), B(–14, – 6)

2) The endpoints of a circle's diameter are at A(2, 3) and B(6, -5).

a) Find the coordinates of the center of the circle.(hint: the center is the midpoint)

b) What is the measure of the radius? (hint: the distance from the midpoint to A or B)

3) A cargo boat leaves Port A (–30, – 40) heading for Port B (90, 60) but has engine trouble when it reaches P at of the distance from A to B. Another boat located at C (–10, 35) must reach the cargo boat in order to make repairs on the engine.

How far must the repair boat travel to meet up with the cargo boat? (scale in kilometers)

Solutions

1) In what ratio does point P divide segment AB? (make a diagram)

a) P(2, 0), A(–2, – 6), B(6, 6) P is the midpoint of AB so the ratio is 1:1	b) P(0, 0), A(8, – 6), B(–4, 3) the ratio is 2 : 1
c) P( , 5), A(2, 7), B(–1, – 5) the ratio is 1 : 5	d) P(–7, – 4), A(28, 6), B(–14, – 6) the ratio is 5 : 1

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2) The endpoints of a circle's diameter are at A(2, 3) and B(6, – 5).

a) Find the coordinates of the center of the circle.(hint: the center is the midpoint)

the midpoint is (4, 1)

b) What is the measure of the radius? (hint: the distance from the midpoint to A or B)

the radius = .

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3)

First we find the coordinates of P, then we find the distance from C to P.

Since AP is of AB, AD must be of AE.

AE = 90 – (– 30) = 120, so AD = (120) = 75

Similarly, DP is of EB.

EB = 60 – (– 40) = 100, so EB = (100) = 62.5

We add 75 to the initial x-value of – 30 to get 45, the x-coordinate of P.

We add 62.5 to the initial y-value of – 40 to get 22.5, the y-coordinate of P.

Point P is at (45, 22.5)

The distance CP = km.

The repair boat must travel 56.4 kilometers to reach the cargo boat.

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Analytic Geometry MathRoom Index

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