Math Lesson 14.5.5 - Coordinates and Ratio

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

Welcome to our Math lesson on Coordinates and Ratio, this is the fifth lesson of our suite of math lessons covering the topic of Line Segments, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Coordinates and Ratio

We can use ratios to identify and express the position of a given point in a line segment. The rules are the same as those explained in the fourth chapter of this course. For example, if we say the AP : PB ratio in the line segment AB is 3:4, then we conclude that point P is at

of the way from A to B. This clue helps identify the coordinates of point P if the coordinates of points A and B are known. Obviously, the ratio rules are applied for each direction separately to find each of the coordinates of point P. Let's use an example to clarify this point.

Example 5

Find the coordinates of point P, which is a point of the line segment AB with endpoints coordinates A(-1, 3) and B(8, -12), if these three points have the following relationship between them:

Then, confirm the result obtained by plotting the corresponding figure.

Solution 5

From the definition of ratios, we can write

where k is a constant of proportionality. Thus, we can write

Since AB = AP + BP, we have AB = 1k + 2k = 3k. From here, we find that

The component lengths of AB are:

and

Thus, point P will be at 1/3 of the line segment AB away from the leftmost endpoint A. This means the x-coordinate of point P is

on the tight of the endpoint A and

i.e. 5 units below the endpoint A (as AB_y is negative).

Therefore, the point P has the coordinates

and

Therefore, point A has the coordinates P(2, -2).

The figure below shows the line segment AB and the three given points: A, P and B.

We can also find a missing endpoint of a line segment when the other endpoint and the coordinates of a point inside the segment - which is in a given ratio from the two endpoints, - are given. Let's see an example in this regard.

Example 6

Point P lies on the line segment AB. Find the coordinates of B given that:

1. A(-6, 4); P(12, -2); AP:PB = 3:2
2. A(-1, -6); P(4, 9); ST:TU = 5:4

Solution 6

1. First, we find the length of AP in each direction. Thus
   AP_x = x_P - x_A
   = 12 - (-6)
   = 18
   and
   AP_y = y_P - y_A
   = -2 - 4
   = -6
   Given that AP: PB = 3: 2, we have AP = 3k and PB = 2k, so AB = 3k + 2k = 5k.
   Now, let's find the constant of proportionality k in each direction. Given that
   AP_x = 3k_x = 18
   and
   AP_y = 3k_y = -6
   then,
   k_x = 18/3 = 6
   and
   k_y = -6/3 = -2
   Therefore, given that AB = 5k, then
   AB_x = 5k_x
   = 5 ∙ 6
   = 30
   and
   AB_y = 5k_y
   = 5 ∙ (-2)
   = -10
   The coordinates of point B are found by adding those of point A and the length segments AB in each direction. Thus,
   
   x_B = x_A + AB_x
   = -6 + 30
   = 24
   and
   y_B = y_A + AB_y
   = 4 + (-10)
   = -6
   Therefore, the coordinates of the endpoint B are B(24, -6).
2. First, we find the length of AP in each direction. Thus
   AP_x = x_P - x_A
   = 4 - (-1)
   = 4 + 1
   = 5
   and
   AP_y = y_P - y_A
   = 9 - (-6)
   = 9 + 6
   = 15
   Given that AP: PB = 5: 4, we have AP = 5k and PB = 4k, so AB = 5k + 4k = 9k.
   Now, let's find the constant of proportionality k in each direction. Given that
   AP_x = 5k_x = 5
   and
   AP_y = 5k_y = 15
   then
   k_x = 5/5 = 1
   and
   k_y = 17/5 = 3
   Therefore, given that AB = 9k, then
   AB_x = 9k_x
   = 9 ∙ 1
   = 9
   and
   AB_y = 5k_y
   = 9 ∙ 3
   = 27
   The coordinates of point B are found by adding those of point A and the length segments AB in each direction. Thus,
   x_B = x_A + AB_x
   = -1 + 9
   = 8
   and
   y_B = y_A + AB_y
   = -6 + 27
   = 21
   Therefore, the coordinates of the endpoint B are B(8, 21).

More Line Segments Lessons and Learning Resources

Whats next?

Enjoy the "Coordinates and Ratio" math lesson? People who liked the "Line Segments lesson found the following resources useful:

1. Ratio Coordinates Feedback. Helps other - Leave a rating for this ratio coordinates (see below)
2. Linear Graphs Math tutorial: Line Segments. Read the Line Segments math tutorial and build your math knowledge of Linear Graphs
3. Linear Graphs Revision Notes: Line Segments. Print the notes so you can revise the key points covered in the math tutorial for Line Segments
4. Linear Graphs Practice Questions: Line Segments. Test and improve your knowledge of Line Segments with example questins and answers
5. Check your calculations for Linear Graphs questions with our excellent Linear Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Linear Graphs Calculators by iCalculator™ below.
6. Continuing learning linear graphs - read our next math tutorial: Linear Graphs

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

We hope you found this Math tutorial "Line Segments" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Linear Graphs Calculators by iCalculator™