Geometry Warm-Up: Solving Equations and Finding Segment Lengths

Warm Up Solve each equation. 1.2x – 6 = 7x – 31 2.1/4 x – 6 = 220 3.3(5 – 2x) = -3x 2 904 5

Objectives Use length and midpoint of a segment.

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

A B AB = |a – b| or |b - a| a b 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.

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. b. XZ a. XY

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

– 6 –6 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 5 = GH

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

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

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

– 3x – 3x 12 3x = 3 3 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 12 = 3x 4 = x

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

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.

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

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

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

S is the mdpt. of RT. +3x +3x 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 –2x = –3x – 2 Substitute –2x for RS and –3x – 2 for ST. 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. RT = RS + ST RS = –2x ST = –3x – 2 = –2(–2) = –3(–2) – 2 = 4 + 4 = 4 = 4 = 8

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).

Step 2 Use the Midpoint Formula: 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).

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

Find FG and JK. Then determine whether FG  JK. Example 3: Using the Distance Formula 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 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 K (-2, 5) L (1, 1) M (-3, 1)

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?

Geometry Warm-Up: Solving Equations and Finding Segment Lengths