# 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:

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

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

### 2.

Explain to Malik why the logic that made him think the midpoint was

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

### 3.

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

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

### 6.

Use either strategy to find the midpoint of the segment between

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

### 7.

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

The stake should be located at:

There’s only one more set of stakes to put in. This time the endpoint stakes are at

### 8.

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

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

## Takeaways

To find the midpoint of the segment whose endpoints are

To find the point that divides the segment whose endpoints are

Strategy:

Rule:

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

## Vocabulary

- directed distance
- midpoint
- ratio
**Bold**terms are new in this lesson.

## 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.

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.

Use the slope and point provided to write the equation of the line in slope-intercept form.

Slope: