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Outline • Collect syllabi • Go over daily quiz • Answer homework questions (markings?) • Daily Quiz • Lecture 1.3
Outline• Collect syllabi
• Go over daily quiz
• Answer homework questions (markings?)
• Daily Quiz
• Lecture 1.3
Front Side – True or False
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?
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 IA B
So….. AM = MB
M
Segment Bisector
A point, segment, line, or plane that divides a line segment into two equal parts
I I I I I I
In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.
Skateboard
SOLUTION
EXAMPLE 1 Find segment lengths
Point T is the midpoint of XY . So, XT = TY = 39.9 cm.
XY = XT + TY= 39.9 + 39.9= 79.8 cm
Segment Addition PostulateSubstitute.
Add.
Bisect: to cut in 1/2
SOLUTION
EXAMPLE 2 Use algebra with segment lengths
STEP 1 Write and solve an equation. Use the fact that that VM = MW.
VM = MW4x – 1 = 3x + 3
x – 1 = 3x = 4
Write equation.
Substitute.
Subtract 3x from each side.Add 1 to each side.
Point M is the midpoint of VW . Find the length of VM .ALGEBRA
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.
MW = 3x + 3 = 3(4) +3 = 15
GUIDED PRACTICE
Identify the segment bisector of .
PQ
Then find PQ.
line l
MIDPOINT FORMULA
The midpoint of two points P(x1, y1) and Q(x2, y2) is
M(X,Y) = M(x1 + x2, x2 +y2)
Think of it as taking the average of the x’s and the average of the y’s to make a new point.
2 2
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
252
1 + 4 2
– 3 + 2 2 =, M , – 1M
The coordinates of the midpoint M are 1,–5
2 2
ANSWER
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
EXAMPLE 3 Use the Midpoint Formula
FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.
STEP 1 Find x.
1+ x 22
=
1 + x = 4
x = 3
STEP 2 Find y.
4+ y 12
=
4 + y = 2
y = – 2
The coordinates of endpoint K are (3, – 2).ANSWER
b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
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
SOLUTION
EXAMPLE 4 Standardized Test Practice
Use the Distance Formula. You may find it helpful to draw a diagram.
EXAMPLE 4 Standardized Test Practice
Distance Formula
Substitute.
Subtract.
Evaluate powers.
Add.
Use a calculator to approximate the square root.
(x – x ) + (y – y )2 2 2 2 1 1 RS =
[(4 – 2)] + [(–1) –3] 2 2=
(2) + (–4 )2 2=
4+16=
20=
4.47=
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?
