Warm Up Solve each equation. 1. 2x – 6 = 7x – 31 2. 1/4 x – 6 = 220 904 5

Objectives Use length and midpoint of a segment.

Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments

The distance between any two points is the absolute value of the difference of the coordinates. The distance between A and B is also called the length of AB, or AB. AB = |a – b| or |b - a| A a B b

Example 1: Finding the Length of a Segment Find each length. A. BC B. AC BC = |1 – 3| AC = |–2 – 3| = |1 – 3| = |– 5| = 2 = 5

Check It Out! Example 1 Find each length. a. XY b. XZ

In order for you to say that a point B is between two points A and C, all three points must lie on the same line – collinear. AB + BC = AC

Example 3A: Using the Segment Addition Postulate G is between F and H, FG = 6, and FH = 11. Find GH. Hint: First draw the diagram. FH = FG + GH 11 = 6 + GH – 6 –6 5 = GH

Example 3a Y is between X and Z, XZ = 3, and XY = . Find YZ. XZ = XY + YZ

TRY THIS… M is between N and O. Find NO. NM + MO = NO 17 + (3x – 5) = 5x + 2 3x + 12 = 5x + 2 – 2 – 2 3x + 10 = 5x –3x –3x 10 = 2x 2 5 = x

NO = 5x + 2 = 5(5) + 2 = 27 Check Your Work!!!!!!! M is between N and O. Find NO. NO = 5x + 2 = 5(5) + 2 Substitute 5 for x. = 27 Simplify.

Check It Out! Example 3b E is between D and F. Find DF. DE + EF = DF (3x – 1) + 13 = 6x Substitute the given values 3x + 12 = 6x – 3x – 3x 12 = 3x 12 3x 3 = 4 = x

Check Your Work! E is between D and F. Find DF. DF = 6x = 6(4) Substitute 4 for x. = 24 Simplify.

Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write PQ  RS. “Segment PQ is congruent to segment RS.” Tick marks are used in a figure to show congruent segments.

If M is the midpoint of AB, then AM = MB. The midpoint (middle point) of AB is the point that bisects (divides), the segment into two congruent segments. A M B If M is the midpoint of AB, then AM = MB. So if AB = 6, then AM = 3 and MB = 3

Example 5: Using Midpoints to Find Lengths D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. E D 4x + 6 7x – 9 F Step 1 Solve for x. ED = DF 4x + 6 = 7x – 9 –4x –4x 6 = 3x – 9 +9 + 9 15 = 3x x = 5

ED = 4x + 6 DF = 7x – 9 EF = ED + DF = 4(5) + 6 = 7(5) – 9 = 26 + 26 Always Check Your Work!!!! D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. E D 4x + 6 7x – 9 F Step 2 Find ED, DF, and EF. ED = 4x + 6 DF = 7x – 9 EF = ED + DF = 4(5) + 6 = 7(5) – 9 = 26 + 26 = 26 = 52 = 26

RS = ST –2x = –3x – 2 +3x +3x x = –2 Can x be negative? DO NOW S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R S T –2x –3x – 2 Step 1 Solve for x. RS = ST S is the mdpt. of RT. –2x = –3x – 2 Substitute –2x for RS and –3x – 2 for ST. +3x +3x x = –2 Can x be negative?

Are you checking your work????? S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R S T –2x –3x – 2 Step 2 Find RS, ST, and RT. RS = –2x ST = –3x – 2 RT = RS + ST = –2(–2) = –3(–2) – 2 = 4 + 4 = 4 = 4 = 8

1. M is between N and O. MO = 15, and MN = 7.6. Find NO. Lesson Quiz: Part I 1. M is between N and O. MO = 15, and MN = 7.6. Find NO. 22.6 2. S is the midpoint of TV, TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV. 25, 25, 50 3. LH bisects GK at M. GM = 2x + 6, and GK = 24. Find x. 3

Round Table

Independent Practice P. 12 #6 – 12 P. 19 # 5, 11 – 13

Do Now Quick Write Can a line have a midpoint or bisector? Explain? What is the difference between a point on a line, and the midpoint? How does this difference affect how you set up an equation for each type of problem? }

Objective SWBAT calculate the length and midpoint of a segment in a coordinate plane.

Line Segments in a coordinate plane Vertices Midpoint Endpoints Length Could you derive a formula to find the midpoint of this line segment?

Midpoint in a Coordinate Plane The midpoint M of AB with endpoints A( X1, Y1) and B(X2 , Y2) is found by:

Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). = (–5, 5)

Check It Out! Example 1 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).

EXTENSION: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y). Step 2 Use the Midpoint Formula:

Step 3 Find the x-coordinate. The coordinates of Y are (10, –5). Example 2 Continued Step 3 Find the x-coordinate. 12 = 2 + x 2 = 7 + y – 7 –7 – 2 –2 –5 = y 10 = x The coordinates of Y are (10, –5).

Example 3: Using the Distance Formula Find FG and JK. Then determine whether FG  JK. Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)

Find EF and GH. Then determine if EF  GH. Check It Out! Example 3 Find EF and GH. Then determine if EF  GH. Step 1 Find the coordinates of each point. E(–2, 1), F(–5, 5), G(–1, –2), H(3, 1)

Check It Out! Example 3 Continued Step 2 Use the Distance Formula.

Find the perimeter of Triangle KLM

Independent Practice P. 19 #17 – 22 Challenge P.20 #26 & 27

Exit Ticket Find the midpoint of AB when A(6, -2) and B(8, -5) Challenge Exit Ticket Given line segment AB where A(5, 4) and midpoint M(3, 3). What are the coordinates of B?