1-2 Measuring and Constructing Segments Warm Up Solve each equation. 1. 2x – 6 = 7x – 31 2 2. 1/4 x – 6 = 220 904 3. 3(5 – 2x) = -3x 5 Holt McDougal Geometry 1-2 Measuring and Constructing Segments Objectives Use length and midpoint of a segment. Holt McDougal Geometry 1-2 Measuring and Constructing Segments Vocabulary coordinate midpoint distance bisect length segment bisector construction between congruent segments Holt McDougal Geometry 1-2 Measuring and Constructing 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. A a Holt McDougal Geometry B b AB = |a – b| or |b - a| 1-2 Measuring and Constructing Segments 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 Holt McDougal Geometry 1-2 Measuring and Constructing Segments Check It Out! Example 1 Find each length. a. XY Holt McDougal Geometry b. XZ 1-2 Measuring and Constructing Segments 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 Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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 Holt McDougal Geometry 1-2 Measuring and Constructing Segments Example 3a Y is between X and Z, XZ = 3, and XY = Find YZ. XZ = XY + YZ Holt McDougal Geometry . 1-2 Measuring and Constructing Segments 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 2 5=x Holt McDougal Geometry 1-2 Measuring and Constructing Segments Check Your Work!!!!!!! M is between N and O. Find NO. NO = 5x + 2 = 5(5) + 2 = 27 Holt McDougal Geometry Substitute 5 for x. Simplify. 1-2 Measuring and Constructing Segments Check It Out! Example 3b E is between D and F. Find DF. DE + EF = DF (3x – 1) + 13 = 6x 3x + 12 = 6x – 3x – 3x 12 = 3x 12 3x = 3 3 4=x Holt McDougal Geometry Substitute the given values 1-2 Measuring and Constructing Segments Check Your Work! E is between D and F. Find DF. DF = 6x = 6(4) Substitute 4 for x. = 24 Simplify. Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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. Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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 Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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 4x + 6 D 7x – 9 Step 1 Solve for x. ED = DF 4x + 6 = 7x – 9 –4x –4x 6 = 3x – 9 +9 +9 15 = 3x x=5 Holt McDougal Geometry F 1-2 Measuring and Constructing Segments Always Check Your Work!!!! D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. E 4x + 6 D 7x – 9 F Step 2 Find ED, DF, and EF. DF = 7x – 9 ED = 4x + 6 = 7(5) – 9 = 4(5) + 6 = 26 = 26 Holt McDougal Geometry EF = ED + DF = 26 + 26 = 52 1-2 Measuring and Constructing Segments DO NOW S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R –2x S –3x – 2 T Step 1 Solve for x. S is the mdpt. of RT. RS = ST –2x = –3x – 2 Substitute –2x for RS and –3x – 2 for ST. +3x +3x x = –2 Holt McDougal Geometry Can x be negative? 1-2 Measuring and Constructing Segments Are you checking your work????? S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. R –2x S –3x – 2 T Step 2 Find RS, ST, and RT. RS = –2x = –2(–2) =4 Holt McDougal Geometry ST = –3x – 2 = –3(–2) – 2 =4 RT = RS + ST = 4 + 4 =8 1-2 Measuring and Constructing Segments 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 Holt McDougal Geometry 1-2 Measuring and Constructing Segments Round Table Holt McDougal Geometry 1-2 Measuring and Constructing Segments Independent Practice P. 12 #6 – 12 P. 19 # 5, 11 – 13 Holt McDougal Geometry 1-2 Measuring and Constructing Segments Do Now Quick Write 1. Can a line have a midpoint or bisector? Explain? 2. 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? } Holt McDougal Geometry 1-2 Measuring and Constructing Segments Objective • SWBAT calculate the length and midpoint of a segment in a coordinate plane. Holt McDougal Geometry 1-2 Measuring and Constructing Segments Line Segments in a coordinate plane • Vertices • Midpoint • Endpoints • Length Could you derive a formula to find the midpoint of this line segment? Holt McDougal Geometry 1-2 Measuring and Constructing Segments Midpoint in a Coordinate Plane The midpoint M of AB with endpoints A( X1, Y1) and B(X2 , Y2) is found by: Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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) Holt McDougal Geometry 1-2 Measuring and Constructing Segments Check It Out! Example 1 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3). Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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: Holt McDougal Geometry 1-2 Measuring and Constructing Segments Example 2 Continued Step 3 Find the x-coordinate. 12 = 2 + x – –2 2 10 = x The coordinates of Y are (10, –5). Holt McDougal Geometry 2=7+y – –7 7 –5 = y 1-2 Measuring and Constructing Segments Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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) Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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) Holt McDougal Geometry 1-2 Measuring and Constructing Segments Check It Out! Example 3 Continued Step 2 Use the Distance Formula. Holt McDougal Geometry 1-2 Measuring and Constructing Segments Find the perimeter of Triangle KLM K (-2, 5) L (1, 1) M (-3, 1) Holt McDougal Geometry 1-2 Measuring and Constructing Segments Independent Practice P. 19 #17 – 22 Challenge P.20 #26 & 27 Holt McDougal Geometry 1-2 Measuring and Constructing Segments 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? Holt McDougal Geometry