Geometry Unit1: Segment Relationships Notes Unit 1 ...

Geometry Unit1: Segment Relationships Notes

1

Unit 1: Segment Relationships After completion of this unit, you will be able to…

• Name lines, rays, and segments

• Define and recognize the following relationships

o Midpoint

o Segment Addition

o Segment Bisector

• Use the relationships to find missing segment lengths

Timeline for Unit 1

Monday Tuesday Wednesday Thursday Friday

26

Visualization

Practice

27

Day 1

Naming Points,

Lines, Rays, and

Segments

28

Day 2

Segment

Relationships

29

Day 2

Segment

Relationships

30

Day 3

Distance Formula

2

No School

3

Day 4

Midpoint Formula

4

Review

5

Unit 1 Assessment

6


2

Day 1 – Naming Points, Lines, Segments, and Rays Notes

Term Definition Picture Name

Point* A point indicates location

and has no size.

Line*

A line is represented by a

straight path that extends in

opposite directions without

an end.

Plane*

A plane is represented by a

flat surface that extends

without end.

Segment

A segment is part of a line

that consists of two

endpoints and all the points

between them.

Ray

A ray is part of a line that

consists of one endpoint

and extends infinitely in the

other direction.

Opposite Rays

Opposite rays are two rays

that share the same

endpoint and form a line.

Collinear Points on the same line. N/A

Coplanar Points and lines that lie in

the same plane. N/A

*Undefined Term: Basic idea you use to build definitions for other figures


3

A postulate is an accepted statement of fact.

Postulate 1: Through any two points, there is

exactly one line.

Postulate 2: If two distinct lines intersect, they

intersect in exactly one point.

Postulate 3: If two distinct planes intersect, then

they intersect in exactly one line.

Postulate 4: Through any three noncollinear

points, there is exactly one plane.

Name the following in as many ways as possible:

1. 2. 3.

4. Use the figure at the right to answer the following questions:

a. Are points F, B, and W collinear?

b. Name the plane.

c. Name 4 coplanar points.

d. At what point does line ℓ and 𝐵𝐼↔ intersect?


4

Types of Lines

Draw and label a figure for each relationship:

a. Point K lies on 𝑅𝑇 ⃡ b. 𝐴𝐵and 𝐶𝐷 intersect at point E.

Day 1 – Naming Points, Lines, and Planes Practice

1. Write the following using correct symbols:

2. Determine if the following are true or false.

Parallel

𝑨𝑩 ⃡ ∥ 𝑪𝑫 ⃡

Lines that do not intersect.

Intersecting

intersectAB CD

Lines that intersect at a

point.

Perpendicular

AB CD⊥ Lines that intersect at a 90

degree angle.


5

3. Find the intersection of:

4. Determine the following:

5. Name the following:

a. b. c.

d. e. f.


6

Day 2: Segment Relationships Notes

Think About It: See if you can determine the length of the requested segment. Draw a picture first.

1. C is between A and E. AC = 24 in and CE = 13 in. How long is AE?

2. C is between A and E. CE = 7 in and AE = 23 in. How long is AC?

Segment Addition Postulate: If point B is on𝐴𝐶, and between points A and C, then 𝐴𝐵 + 𝐵𝐶 =𝐴𝐶.

a. Use the diagram to find𝐸𝐹.

b. Write an expression for AC. c. Find the value of z.

Segment Addition

Given:

Conclusion:


7

Midpoint: Point that divides the segment into two congruent segments.

a. Find 𝐹𝑀 and 𝑀𝐺. b. Find 𝑌𝑀and𝑌𝑍.

c. T is the midpoint of𝑄𝑅. Solve for x.

Segment Bisector: A line, line segment, or ray that divides the line segment into two line segments of equal

length.

a. Find 𝐶𝐵 and 𝐴𝐵.

b. Determine if you have enough information to determine if 𝑃𝐶is the

segment bisector of 𝐴𝐵. Explain why or why not.

Midpoint

Given:

Conclusion:

Segment Bisector

Given:

Conclusion:


8

Day 2: Segment Relationships Practice

Find the measure of the stated segment.

1. 2. 3.

4. 5. 6.

7. If RS = TU, ST = 19, RU = 33

a) Find RS

b) Find SU.

8. In the diagram, points V, W, X, Y, and Z are collinear. VZ = 52, XZ= 20, and WX = XY = YZ. Find the indicated

lengths.

a. WX d. VW

b. VX e. WZ

c. WY f. VY

9. M is the midpoint of JL. Find JM. 10. Line l is the segment bisector of PQ . Solve for x.


9

11. Solve for x. Then determine the length of AC. 12. How long is BC?

13. If U is between T and B, TU = 4x – 1, UB = 2x – 1, and TB = 5x, determine:

a. x = ______

b. TU = ______

c. UB = ______

d. TB = ______


10

Day 3 – Distance Formula

How would you find the distance between the coins and beads?

The Distance Formula allows you to find the distance between two points. The subscripts (x1, y1) only indicate

that there is a first and second point. However, whichever point is first or second is up to you.

Distance Formula: 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

1. Find the distance between (1, -2) and (-3, 6). 2. Find the distance between (-2, -3) & (-4, 4).

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


11

Day 3: Distance Formula Practice

Problem 1: Find the distance between the following points. Round to the nearest tenth.

a. (-4, 2) and (2,-1) b. (-2, -3) and (-2, 4) c. (3, 2) and (5, -2)

Problem 2

• Your house is located 3 blocks east and 4 blocks north.

• Town Center Mall is located 1 block west and 5 blocks north.

• Six Flags is centered at the 5 blocks west and 1 block south.

• Iowa Aquarium is located 6 blocks east and 4 blocks south.

A. Plot your house. Label it was Point A. What is the ordered

pair? _______

B. Plot Town Center Mall. Label it as point B. What is the

ordered pair? _________

C. Plot the Six Flags. Label it as point C. What is the ordered

pair? _________

D. Plot Iowa Aquarium. Label it as point D. What is the ordered pair? _________

E. Connect all four points.

Using the distance formula, find the distance (show all your work). Round to the nearest tenth.

F. AB G. BC H. CD I. AD

How many blocks did you travel all together if you left your house to go to the mall, then Six Flags, and then the

Iowa Aquarium?

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


12

Day 4: Midpoint Formula Notes

How would you find the midpoint between the following points?

a. Midpoint of 𝐴𝐷 = _____ b. Midpoint of 𝐴𝐷 = _____

c. Midpoint of 𝐶𝐹 = _____ d. Midpoint of 𝐹𝐵 = _____

The Midpoint Formula allows you to find the midpoint or center between two points.

Midpoint Formula: (𝑥1+𝑥2

2,𝑦1+𝑦2

2)

1. Find the midpoint between (1, -2) and (-3, 6).

2. Find the midpoint between (6.4, 3) and (-10.7, 4).

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


13

3. M is the midpoint of segment AB. The coordinates of A are (-2, 3) and the coordinates of M are (1, 0). Find

the coordinates of B.

4. B is the midpoint of segment AC. The coordinates of A are (-10, 4) and the coordinates of B are (-2,4). Find

the coordinates of C.

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8


14

Day 4: Midpoint Formula Practice

Problem 1: Troop 175 is designing their new campground by first

mapping everything on a coordinate grid. They have found a

location for the mess hall and for their cabins. They want the

bathrooms to be halfway between these two. What will be the

coordinates of the location of the bathrooms?

Problem 2: You and a friend go hiking. You hike 3 miles north and 2 miles

west. Starting from the same point, your friend hikes 4 miles east and 1 mile

south. If you and your friend wanted to meet for lunch, where could you

meet so that both of you hike the same distance?

sProblem 3: Determine the other endpoint given one of the endpoints and the midpoint.

a. Endpoint (0, 3) and Midpoint (3, 5) b. Endpoint (-2, 5) and Midpoint (4, -1)

Problem 4: Points P(-4, 6), Q(2, 4) and R are collinear. One of the points is

the midpoint of the segment formed by the other two points. What are the

possible coordinates of R?

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