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# Midpoints and Segment Bisectors

## Midpoints and bisectors find the halfway mark.

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Practice Midpoints and Segment Bisectors
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Midpoint And Segment Bisectors - Treasure Hunt Activity
Teacher Contributed

## Treasure Hunt

### The Midpoint Formula

#### OBJECTIVES

In this lesson you will:

• Derive the Midpoint Formula.
• Use the Midpoint Formula.

#### KEY TERMS

• midpoint
• Midpoint Formula

Ms. Lopez is designing a treasure hunt for kindergarten children. The goal of the treasure hunt is for children to learn direction (right, left, forward, and backward) and introduce distance (near, far, in between). The treasure hunt will be on the school playground.

PROBLEM 1

Plotting the Treasure Hunt

Ms. Lopez drew a model of the playground on a grid as shown. She uses this model to decide where to place items for the treasure hunt, and to determine how to write the treasure hunt instructions. Each grid square represents one square foot on the playground.

1. What are the coordinates of:
a. the merry-go-round
The merry-go-round is located at (11, 2).
b. the slide
The slide is located at (3, 2).
c. the swings
The swings are located at (3, 12).
2. Determine the distance, in feet, between the merry-go-round and the slide. Show all your work.
11 3 8
The merry-go-round and the slide are eight feet apart.
3. Ms. Lopez wants to place a small pile of beads in the grass halfway between the merry-go-round and the slide. How far, in feet, from the merry-go-round should the beads be placed? How far, in feet, from the slide should the beads be placed?
The beads should be placed four feet from the slide and four feet from the merry-go-round.
4. What should be the coordinates of the pile of beads? Explain how you determined your answer. Plot and label the pile of beads on the coordinate plane for Questions 1 through 4.
The pile of beads will be located four feet to the right of the slide, so its coordinates will be (7, 2).
5. How do the coordinates of the pile of beads compare to the coordinates of the slide and merry-go-round?
The $y-$coordinates of all three items are the same. The $x-$coordinate of the pile of beads is 4 units greater than the $x-$coordinate of the slide and is 4 units less than the $x-$coordinate of the merry-go-round.
6. Ms. Lopez also wants to place a pile of kazoos in the grass halfway between the slide and the swings. What should the coordinates of the pile of kazoos be?
Explain your reasoning. Plot and label the pile of kazoos on the grid.
Distance between swings and slide: 12 2 10
Distance between slide and pile of kazoos: 5
The pile of kazoos will be 5 feet greater on the $y-$coordinate than the slide, so its coordinates will be (3, 7).
7. How do the coordinates of the pile of kazoos compare to the coordinates of the slide and swings?
The $x-$coordinates of all three items are the same. The $y-$coordinate of the pile of kazoos is 5 units greater than the $y-$coordinate of the slide and is 5 units less than the $y-$coordinate of the swings.
8. Ms. Lopez wants to place a pile of buttons in the grass halfway between the swings and the merry-go-round. What do you think the coordinates of the pile of buttons will be? Explain your reasoning. Plot and label the pile of buttons on the shown grid.
The coordinates of the pile of buttons should be (7, 7). The $x-$coordinate is halfway between the $x-$coordinates of the slide and merry-go-round, and the $y-$coordinate is halfway between the $y-$coordinates of the swings and slide.
9. How far, in feet, from the swings and the merry-go-round will the pile of buttons be? Show all your work and explain how you determined your answer.
Let $d$ be the distance between the swings and the merry-go-round.
$(3 \quad 11)^2 \quad (12 \quad 2)^2 \quad d^2\\( \ 8)^2 \quad 10^2 \quad d^2\\64 \quad 100 \quad d^2\\164 \quad d^2\\12.8 \quad d$