Lesson 2 Yard Work in Segments Solidify Understanding

Learning Focus

Locate the midpoint of a segment and a point that divides the segment in a given ratio.

How do I locate the midpoint of a segment given just the coordinates of its endpoints?

How do I divide a line segment drawn on a grid into proportional parts?

Open Up the Math: Launch, Explore, Discuss

Malik’s family has purchased a new house with an unfinished yard. They drew the following map of the backyard:

irregular hexagon with line segments ((-9,6) to (3,10)), ((1,5) to (4,11)), and ((-8,5) to (2,-10)) in the interior. x–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414y–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212141414000(4, 11)(4, 11)(4, 11)(1, 5)(1, 5)(1, 5)(3, 10)(3, 10)(3, 10)(-9, 6)(-9, 6)(-9, 6)(-8, 5)(-8, 5)(-8, 5)(2, -10)(2, -10)(2, -10)

Malik and his family are using the map to set up gardens and patios for the yard. They plan to lay out the yard with stakes and strings (shown as points and dotted lines in the map) so they know where to plant grass, flowers, or vegetables. They want to begin with a vegetable garden that will be parallel to the fence shown at the top of the map.

1.

They set the first stake at and the stake at the end of the garden at . They want to mark the middle of the garden with another stake. Where should the stake that is at the midpoint of the segment between the two end stakes be located? Using a diagram, describe your strategy for finding this point.

Line segment from (-9,6) to (3,10). x–10–10–10–5–5–5555101010y555101010000(3, 10)(3, 10)(3, 10)(-9, 6)(-9, 6)(-9, 6)

Midpoint:

Malik figured out the midpoint by saying, “It makes sense to me that the midpoint is going to be halfway over and halfway up, so I drew a right triangle and cut the horizontal side in half and the vertical side in half like this:”

Malik continued, “That put me right at . The only thing that seems funny about that to me is that I know the base of the big triangle was and the height of the triangle was , so I thought the midpoint might be .

Line segment from (-9,6) to (3,10) broken into triangles representing slope. x–10–10–10–5–5–5555y555101010000

2.

Explain to Malik why the logic that made him think the midpoint was is almost right, and how to extend his thinking to use the coordinates of the endpoints to get the midpoint of .

Malik’s sister, Sapana, looked at his drawing and said, “Hey, I drew the same picture, but I noticed the two smaller triangles that were formed were congruent. Since I didn’t know for sure what the midpoint was, I called it . Then I used that point to write an expression for the length of the sides of the small triangles. For instance, I figured that the base of the lower triangle was .

3.

Label all of the other legs of the two smaller right triangles using Sapana’s strategy.

Line segment from (-9,6) to (3,10) broken into triangles representing slope. x–10–10–10–5–5–5555y555101010000A = (-9, 6)A = (-9, 6)A = (-9, 6)B = (3, 10)B = (3, 10)B = (3, 10)(x, y)(x, y)(x, y)

Sapana continued, “Once I labeled the triangles, I wrote equations by making the bases equal and the heights equal.”

4.

Does Sapana’s strategy work? Show why or why not.

5.

Choose a strategy and use it to find the midpoint of the segment with endpoints and .

6.

Use either strategy to find the midpoint of the segment between and .

Line segment from (x1,y1) to (x2,y2) broken into triangles representing slope. (x(x(x111, y, y, y111)))(x(x(x222, y, y, y222)))

Pause and Reflect

The next area in the garden to be marked is for a flower garden. Malik’s parents have the idea that part of the garden should contain a big rose bush and the rest of the garden will have smaller flowers like petunias. They want the section with the other flowers to be twice as long as the section with the rose bush. The stake on the endpoints of this garden will be at and . Malik’s dad says, “We’ll need a stake that marks the end of the rose garden.”

7.

Help Malik and Sapana figure out where the stake will be located if the rose bush will be closer to the stake at than the stake at .

Line segment from (1,5) to (4,11) x222444666y222444666888101010121212000(1, 5)(1, 5)(1, 5)(4, 11)(4, 11)(4, 11)

The stake should be located at:

There’s only one more set of stakes to put in. This time the endpoint stakes are at and . Another stake needs to be placed that partitions this segment into two parts so that the ratio of the lengths is .

8.

Where must the stake be located if it is to be closer to the stake at than to the stake at ?

Line segment from (-8,5) to (2,-10) x–10–10–10–5–5–5555y–10–10–10–5–5–5555000(2, -10)(2, -10)(2, -10)(-8, 5)(-8, 5)(-8, 5)

The stake should be located at:

Ready for More?

Generate a symbolic rule for locating the point that divides a line segment into two parts so that the ratio of the lengths is , with the point closer to the left endpoint.

Takeaways

To find the midpoint of the segment whose endpoints are and , I can use the midpoint rule:

To find the point that divides the segment whose endpoints are and into two parts in the ratio , I can use a strategic method or remember the rule:

Strategy:

Rule:

Here are my illustration and notes to explain why this rule or strategy works:

Vocabulary

Lesson Summary

In this lesson, we examined strategies for dividing a line segment into two parts that fit a given ratio. One common application of this concept is to find the coordinates of the midpoint of a segment, given the coordinates of the endpoints.

Retrieval

Find the mean, or average, for each set of numbers.

1.

, ,

2.

, , ,

Find the value that is exactly halfway between the two given values.

3.

,

4.

,

5.

Explain your process for finding the value halfway between two values in the last two problems.

6.

Solve the proportion to solve the missing value.