1. Pearson Unit 1 Topic 1: Tools of Geometry 1-2: Measuring Segments Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(2)(A) Determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint.
(1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

3. Virtual Nerd Videos
What Does Congruent Mean?

4. Basic Terms
Two segments are congruent ( ) segments if they have the same length. In the diagram, PQ = RS, so you can write PQ  RS. This is read as “segment PQ is congruent to segment RS.” Tick marks are used in a figure to show congruent segments.

5. Basic Terms
A ruler can be used to measure the distance between two points. A point corresponds to one and only one number on a ruler. The coordinate of a point is the real number that corresponds to a point. P

6. Basic Terms
The distance between any two points is the absolute value of the difference of their coordinates.
Distance is always positive!
If the coordinates of points A and B are a and b, then the distance between A and B is |a – b| or |b – a|. The distance between A and B is also called the length of AB.
AB = | a – b | or |b - a|
Remember: answer is
always POSITIVE. A a B b

7. Basic Terms
The midpoint of a segment is a point that bisects, or divides, the segment into two congruent () segments.
If M is the midpoint of AB, then AM = MB.
So if AB = 6, then AM = 3 and MB = 3.

8. Basic Terms
A segment bisector is a point, a line, a ray, or another segment that intersects a segment at its midpoint (or bisects it).

10. Take 1 minute! A has coordinate of -12 and B has coordinate of 4.
Discuss with a classmate what the
average of A and B would be.
Would the answer for the average
be the coordinate of the midpoint?

14. Example 1:
Find each length.
a. XY
b. XZ

15. Example 2: G is between F and H, FG = 6, and FH = 11. Find GH. 6 11
FG + GH = FH
Seg. Add. Postulate
6 + GH = 11
Substitute 6 for FG and 11 for FH.
– –6
Subtract 6 from both sides.
GH = 5
Simplify.

16. Example 3:
A carpenter has a wooden dowel that is 72 cm long. She wants to cut it into two pieces so that one piece is 5 times as long as the other. What are the lengths of the two pieces?
Let x be length of shorter piece. Let A and B be ends of dowel, and let C be cut point, nearest to A.
AC + CB = 72
x + 5x = 72
6x = 72
x = 12
AC = x = 12
CB = 5x = 5(12) = 60
Dowel pieces are 12 cm long and 60 cm long. A B C x 5x

17. Example 4: M is between N and O. Find NO. NM + MO = NO
Seg. Add. Postulate
17 + (3x – 5) = 5x + 2
Substitute the given values
3x + 12 = 5x + 2
Simplify.
– – 2
Subtract 2 from both sides.
3x + 10 = 5x
Simplify.
–3x –3x
Subtract 3x from both sides.
10 = 2x
Divide both sides by 2. 2
5 = x
Substitute 5 for x and simplify.
NO = 5x + 2
= 5(5) + 2
NO = 27

18. Example 5:
The map shows the route for a race. You are at X, 6000 ft from the first checkpoint C. The second checkpoint D is located at the midpoint between C and the end of the race Y. The total race is 3 miles. How far apart are the 2 checkpoints?
XY = 3(5280 ft)
Convert race distance to feet.
= 15,840 ft

19. Example 5: (continued) Answer: The checkpoints are 4920 ft apart.
XC + CY = XY
Seg. Add. Post.
Substitute 6000 for XC and 15,840 for XY.
CY = 15,840
– – 6000
Subtract 6000 from both sides.
CY = 9840
Simplify.
D is the midpt. of CY, so CD = CY.
= 4920 ft
Answer: The checkpoints are 4920 ft apart.

20. Example 6:
D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. D F E
4x + 6
7x – 9
Step 1 Solve for x.
ED = DF
D is the midpoint. of EF.
4x + 6 = 7x – 9
Substitute 4x + 6 for ED and 7x – 9 for DF.
–4x –4x
Subtract 4x from both sides.
6 = 3x – 9
Simplify.
Add 9 to both sides.
15 = 3x
Simplify.

21. Example 6: (continued)
D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF. D F E
4x + 6
7x – 9 x =
Divide both sides by 3.
Simplify.
x = 5
Step 2 Find ED, DF, and EF.
ED = 4x + 6
DF = 7x – 9
EF = ED + DF
= 4(5) + 6
= 7(5) – 9 =
= 26
= 26
= 52

22. Example: 7
Y is between X and Z, XZ = 3, and XY = Find YZ.
XZ = XY + YZ
Seg. Add. Postulate
Substitute the given values.
Subtract from both sides.

23. Example: 8 E is between D and F. Find DF. DE + EF = DF
Seg. Add. Postulate
(3x – 1) + 13 = 6x
Substitute the given values
3x + 12 = 6x
– 3x – 3x
Subtract 3x from both sides.
12 = 3x
Simplify. x 3 =
Divide both sides by 3.
4 = x
DF = 6x
= 6(4)
= 24
Substitute 4 for x and simplify.

24. Example: 9
You are m from the first-aid station. What is the distance to a drink station located at the midpoint between your current location and the first-aid station?
The distance XY is m. The midpoint would be .

25. Example: 10
S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT. S T R
–2x
–3x – 2
Step 1 Solve for x.
RS = ST
S is the mdpt. of RT.
–2x = –3x – 2
Substitute –2x for RS and –3x – 2 for ST.
+3x +3x
Add 3x to both sides.
x = –2
Simplify.
Step 2 Find RS, ST, and RT.
RS = –2x
ST = –3x – 2
RT = RS + ST
= –2(–2)
= –3(–2) – 2
= 4 + 4
= 4
= 4
= 8