Outline

transcript

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

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.

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 .

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.

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.

1 + 4 2

– 3 + 2 2 =, M , – 1M

The coordinates of the midpoint M are 1,–5

ANSWER

SOLUTION

a. FIND MIDPOINT 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

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

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.

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=

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?

Outline

Documents