[G.GPE.6] Expressing Geometric Properties With Equations #6

Objective

Common Core Text:

• • [G.GPE.6] Find the point on a directed line segment between two given points that partitions the segment in a given ratio

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Explanation

We'll learn this one through an example, and summarize once we're finished

Example 1

You're designing a bike trail from point A to point C. You want it to have two themes. The theme for the first part of the trial will be "woods", with lots of trees. The second theme will be "desert" with lots of rocks and cacti. You decide that the ratio of woods to desert will be 3:2. At what point along the trail will you need to switch from woods to desert?

Our ratio is 3:2 woods to desert. So we need to find the point on this line so that three parts will be woods for every two parts that are desert.

Whenever I see a ratio, my first goal is to convert it to a fraction. Fractions are generally much easier to do math with. So let's convert.

3:2 = 3 parts woods : 2 parts desert

So 3/5 of the trail will be woods and 2/5 of the trail will trail will be desert. We know we did this correctly because 3/5 + 2/5 = 1.

Now that we have a fraction, we can start doing some math. Starting at A, we know we must go 3/5 of the way to B before we switch themes. The easiest way is to take 3/5 of the x-distance and 3/5 of the y-distance. So, how long are these distances? As always, whenever we have a sloped line and we want to know x and y information, we draw a triangle.

I can see that I went from 4 to 12 in the x direction, so the x component is 8. I went from 2 to 8 in the y direction, so the y component is 6.

Well, we only want 3/5 of these distances, so we multiply them by 3/5