1.4 How Can I Find It? Pg. 20 Midpoints and Constructions

1.4 – How Can I Find It?______________ Midpoints and Constructions Today you are going to continue to develop your vocabulary for geometry as well as use your algebra knowledge. You will also use a straightedge and compass to explore different parts of shapes

1.20 – MIDPOINT Segments have a point that can be found in the middle, called a midpoint. This stands for “middle point.” Examine the pictures below. Find the point in the middle of the segment. M M

1.21 – MIDPOINTS ON GRAPHS Find the midpoint of segment AB. Look for any shortcuts that might help you. A(2, 5) M(, ) B(2, -3) a. A(2, 5) and B(2, -3) A B M 21

A(-3, 4) M(, ) B(7, 4) b. A(-3, 4) and B(7, 4) A B M 24

A(5, 8) M(, ) B(1, 2) c. A(5, 8) and B(1, 2) A B M 35

A(-7, 6) M(, ) B(3, -4) d. A(-7, 6) and B(3, -4) A B M -21

1.22 – MIDPOINT FORMULA As Cassie worked on the previous problem, Esther, had difficulty finding the midpoint. The study team decided to try to find another way to find the midpoint of a line segment.

a. Esther thinks she understands how to find the midpoint on the graph. "I always look for the middle of the line segment. But what if the coordinates are not easy to graph?" she asks. With your team, find the midpoint of KL if K(2, 125) and L(98, 15). Be ready to share your method with the class. K(2, 125) L(98, 15) (50,70) ___ 2 2

b. Test your team's method by verifying that the midpoint between (–5, 7) and (9, 4) is (2, 5.5). (–5, 7) ( 9, 4) 411 (2,5.5) ___ 2 2

c. Explain in your own words how to find the midpoint of a segment. Add the x’s and y’s then divide by 2

d. Use the method you described in part (c) to find the midpoint of AB, where A(-6, 2)and B(-2, -3). (–6, 2) (–2, –3) –8 –1 (–4,–0.5) ___ 2 2

1.23 – FINDING AN ENDPOINT As Cassie continued to work with her group, she started to think about a different situation. What if you knew the midpoint, but not the end of one side of the segment. Examine this with the problems below.

a. The midpoint (M) of segment AB is M(2, 1). If the coordinate of B is B(1, 4), what is the coordinate of the point A? A(, ) M(2, 1) B(1, 4) M B A 3–2

b. The midpoint (M) of segment AB is M(- 1, -2). If the coordinate of A is A(4, 4), what is the coordinate of the point B? A(4, 4) M(-1, -2) B(, ) M B A –6–8

c. The midpoint (M) of segment AB is M(6, 5). If the coordinate of B is B(7, 8), what is the coordinate of the point A? A(, ) M( 6, 5) B( 7, 8) 52

d. The midpoint (M) of segment AB is M(1, 0). If the coordinate of A is A(-1, 2), what is the coordinate of the point B? A( -1, 2) M( 1, 0) B(, ) 3–2

1.24 – COPY A SEGMENT Previously we measured segments with a ruler. Today we are going to use a compass to measure and create a "copy" of the segment. This will allow us to be perfectly exact with how long something is, which is not always possible with a ruler.

a. Examine the segment below. Place your point of the compass on one end and the pencil on the other. Notice how this is the exact length of the segment. AB

b. You are going to make a "copy" of the segment in part (a). Follow the steps below. Step 1: With straightedge, draw a segment longer than the one in part (a) to the right. AB

b. You are going to make a "copy" of the segment in part (a). Follow the steps below. Step 2: Measure the length of the segment with the compass, putting the point and pencil on the two ends of the segment, A and B. AB

b. You are going to make a "copy" of the segment in part (a). Follow the steps below. Step 3: Lift the compass without moving the length and place it on the end of the segment you drew to the right. Turn your compass slightly to leave a mark on the segment. AB

b. You are going to make a "copy" of the segment in part (a). Follow the steps below. Step 4: This should be the same length as the segment in part (a). Make sure they look like the same size. AB

c. Make a copy of the segment below using the same steps as above. CD

1.25 – ADDING SEGMENTS Using the length of the two segments above, make a segment with the following lengths. a. Twice the size of CD. CD

a. AB + CD AB

CD

1.26 – MIDPOINT OF A SEGMENT Now you are going to use the compass to find the exact point in the middle of the segment. Step 1: Put the point of your compass on the left endpoint of the segment, A. Open the compass to be more than half the length of the segment. Then leave a half- circle mark. AB

1.26 – MIDPOINT OF A SEGMENT Now you are going to use the compass to find the exact point in the middle of the segment. Step 2: Without changing the size of the compass, turn it around so the point of the compass is on the other end, B. Leave another half-circle mark. AB

Step 3: The two half-circle marks should cross each other at two points, one above the segment, and one below the segment. Take the straightedge and draw a line that connects these two points. AB

Step 4: Where the line you drew in Step 3 crosses the segment, put a point. That is the midpoint. AB M

1.27 – MIDPOINT OF A SEGMENT Find the midpoint of the segment below using the same steps as the previous question. CD

1.27 – MIDPOINT OF A SEGMENT Find the midpoint of the segment below using the same steps as the previous question. CD M

1.28 – MIDPOINT FORMULA Compare your formula you created with the one below? How are they alike? How are they different?