Outline Collect syllabi Go over daily quiz Answer homework questions (markings?) Daily Quiz Lecture 1.3

Front Side – True or False Naming points, lines, rays, and planes Finding intersections

Back Side Use the diagram to answer the questions. 9. Name one pair of opposite rays. ____________ & ____________ Opposite Rays Share the same end point The 2 rays are on the same line They go in opposite directions

Back Side Use the diagram to answer the questions. 10. Name three lines that intersect at C. _____________ _____________

Homework Questions Did you circle the ones you got wrong? Did you come to class prepared to ask questions? If so, which ones did you get wrong? Are you ready for the Daily Quiz?

Daily Quiz 1.2 – Back side Notice that there is no back…. a. Give two other names for b. Name 3 points that are collinear. c. What is the intersection of line a and line XY? d. What is the intersection of plane C and Plane D? e. Give another name for YX f. Name a pair of opposite rays

WARM-UP Directions: Find x. What do you notice about the relationship between segment AB and segment BC?

Use Midpoint and Distance Formulas 1.3 Lesson Use Midpoint and Distance Formulas

Midpoint The midpoint of a segment is a point that divides a segment into 2 congruent segments. I I A B M So….. AM = MB

Segment Bisector A point, segment, line, or plane that divides a line segment into two equal parts I I I I I I

Point T is the midpoint of XY . So, XT = TY = 39.9 cm. EXAMPLE 1 Bisect: to cut in 1/2 Find segment lengths In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY. Skateboard SOLUTION Point T is the midpoint of XY . So, XT = TY = 39.9 cm. XY = XT + TY Segment Addition Postulate = 39.9 + 39.9 Substitute. = 79.8 cm Add.

Use algebra with segment lengths EXAMPLE 2 Use algebra with segment lengths Point M is the midpoint of VW . Find the length of VM . ALGEBRA SOLUTION STEP 1 Write and solve an equation. Use the fact that that VM = MW. VM = MW Write equation. 4x – 1 = 3x + 3 Substitute. x – 1 = 3 Subtract 3x from each side. x = 4 Add 1 to each side.

Use algebra with segment lengths EXAMPLE 2 Use algebra with segment lengths STEP 2 Evaluate the expression for VM when x = 4. VM = 4x – 1 = 4(4) – 1 = 15 So, the length of VM is 15. Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15. I like how these examples are set up so you can ask the class questions about what the next step could be and click to show the next equation or step or reason and things MW = 3x + 3 = 3(4) +3 = 15

GUIDED PRACTICE line l Identify the segment bisector of . Then find PQ.

MIDPOINT FORMULA The midpoint of two points P(x1, y1) and Q(x2, y2) is M(X,Y) = M(x1 + x2, x2 +y2) 2 2 Think of it as taking the average of the x’s and the average of the y’s to make a new point.

EXAMPLE 3 Use the Midpoint Formula a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

EXAMPLE 3 Use the Midpoint Formula SOLUTION a. FIND MIDPOINT Use the Midpoint Formula. 2 5 1 + 4 – 3 + 2 = , M – 1 The coordinates of the midpoint M are 1 , – 5 2 ANSWER

EXAMPLE 3 Use the Midpoint Formula b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K. FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula. STEP 1 Find x. STEP 2 Find y. 1+ x 2 = 4+ y 1 2 = 1 + x = 4 4 + y = 2 x = 3 y = – 2 The coordinates of endpoint K are (3, – 2). ANSWER

Guided Practice A. The endpoints of are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M. B. The midpoint of is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.

Distance Formula The distance between two points A and B is

EXAMPLE 4 Standardized Test Practice SOLUTION Use the Distance Formula. You may find it helpful to draw a diagram.

Standardized Test Practice EXAMPLE 4 Standardized Test Practice (x – x ) + (y – y ) 2 1 RS = Distance Formula [(4 – 2)] + [(–1) –3] 2 = Substitute. (2) + (–4 ) 2 = Subtract. 4+16 = Evaluate powers. 20 = Add. 4.47 = Use a calculator to approximate the square root. The correct answer is C. ANSWER

Example 5 Amy lives 4 blocks north and 6 blocks east of the school. Seth lives 2 blocks south and 7 blocks west of the same school. How far away does Amy live from Seth?