# High School: Geometry

### Expressing Geometric Properties with Equations HSG-GPE.B.6

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Ross and Rachel. Rachel and Ross. Every Friday morning, we gathered around the break room coffee maker to discuss what it would take to split them up. Just like the 18-page wedge that finally came between the two old friends, we can also identify the coordinates of a partition point that splits up a directed line segment.

The three main methods that students can use to find the partition point are as distinct from each other as sitcom character archetypes. Students should be able to use the midpoint formula (if the ratio of the parts is 1:1), the section formula, and the distance formula.

If we're being honest, though, the midpoint formula is just a special case of the section formula. But it comes first in the list because it's the easiest (unlike the game show Pyramid). Just find the averages of the x and y coordinates to find the midpoint, which gives the partition point these coordinates.

The section formula is just a fancier version of the midpoint formula. If a line segment has endpoints (x1, y1) and (x2, y2), and a partition point P will separate the line segment into a ratio of m:n, then students should plug the numbers into the section formula to find the coordinates of P.

Essentially, the midpoint formula is to finding averages as the section formula is to finding weighted averages. Given the endpoints of a line segment, students should be able to use both formulas to find the midpoint M and the partition point P at a specified ratio.

Students should also be able to determine the ratio of a partition using the distance formula. They need to remember that we are talking about directed line segments here. So it does matter on which side of the partition point the bigger segment lies. Remind them to keep track of which segments they're looking at, and hopefully they won't get mad at you. It's not as though you called them boring or anything.

#### Drills

1. What is the midpoint of the line segment with endpoints (-9, -7) and (11, 2)?

(1, -2.5)

Using the midpoint formula to find the averages of the x and y coordinates gives us a midpoint of (1, -2.5). Choice (A) has the coordinates reversed, while (B) and (D) have calculated entered the coordinates into the formula incorrectly.

2. What is the midpoint of the line segment with endpoints (15, -6) and (-13, -1)?

(1, -3.5)

The midpoint formula can be used to find the averages of the x and y coordinates, giving (1, -3.5) as the midpoint. Choice (B) has the signs reversed, while (C) and (D) have entered the endpoint coordinates into the formula incorrectly.

3. What is the midpoint of the line segment with endpoints (0.5, -2.25) and (5.5, -0.75)?

(3, -1.5)

The midpoint formula gives the coordinates (3, -1.5) as the right answer. The other choices reflect errors in executing the formula. It's important to remember that the midpoint is essentially the average of the x and y coordinates, just like the average of a data set gives the approximate midpoint of that data.

4. A line segment has one endpoint (10, 5) and midpoint (16, 7). What is its other endpoint?

(22, 9)

Using the midpoint formula and solving for x2 and y2, we can arrive at the coordinates (22, 9) as the second endpoint. Option (A) is the point halfway between the given endpoint and the midpoint, while (D) is the difference between the two points divided by 2. We're really not sure what (C) is all about.

5. A line segment has one endpoint (-3, -1) and midpoint (-6, 1). What is its other endpoint?

(-9, 3)

The midpoint formula gives us the coordinates of (-9, 3) as the missing endpoint. Choice (A) switches the signs of the coordinates, while (B) calculates the midpoint between these two points and (C) reflects an error in calculation.

6. Line segment AB has endpoints (7, 2) and (4, 6). What are the coordinates of the point that divides AB in the ratio of 2:3?

(5.8, 3.6)

Using the section formula gives us the partition point coordinates of (5.8, 3.6). Make sure to follow the formula exactly! Even switching x1 and x2 changes the answer entirely. Choice (B) forgot to multiply the x and y terms by m and n. Choice (C) combined x and y terms, and (D) subtracted x terms and y terms instead of adding them.

7. Line segment CD has endpoints (-3, 8) and (1, -5). What are the coordinates of the point that divides CD in the ratio of 3:7?

(-1.8, 4.1)

Using the section formula gives us the partition point coordinates of (-1.8, 4). Choice (A) forgot to multiply in m and n and (B) subtracted x and y terms instead of adding them. And (C) was pretty much a random guess.

8. Line segment EF has endpoints (5.5, 2.25) and (6.5, -1.75). What are the coordinates of the point that divides EF in the ratio of 5:3?

(6.125, -0.25)

The section formula yields the coordinates (6.125, -0.25) as the partition point. All we need to do is make sure to follow the formula exactly. Choice (A) is actually the midpoint between these segments. Option (C) didn't multiply the terms by m or n and (D) subtracted the terms instead of adding them.

9. Line segment GH is divided by point I in the ratio of 1:4. Endpoint G is at (7, 5) and point I is at (10, 14). What are the coordinates of endpoint H?

(22, 50)

Here, we can apply the section formula and rearrange it. We know the coordinates of I and G as well as the ratio, so we can solve for x2 and y2. If we rearrange it properly, we should have the endpoint coordinates of (22, 50). You probably selected (A) if you forgot to multiply the x and y terms by m and n. If you chose (B), you probably switched the midpoint coordinates with the given endpoint coordinates in executing the formula. Choice (D) added terms instead of subtracting.

10. Line segment JK is divided by point L in the ratio of 3:2. Endpoint J is at (-16, 2) and point L is at (-15, -20). What are the coordinates of endpoint K?

Use the section formula version of the missing endpoint formula to find the missing endpoint. You probably ended up with (A) if you switched the given endpoint with the midpoint in executing the formula. Adding terms instead of subtracting while executing the formula would have gotten you (B). Option (C) failed to multiply the x and y terms by m and n.