1.2-1.3 Using Segments, Congruence, midpoints and Distance

Learn Segment Postulates so you can identify segment congruence.

Learn midpoint and distance formulas so you can find measurements in coordinate plane

In the study of geometry, definitions , postulates and undefined terms are accepted as true w/o verification or proof.

These 3 types of statements can be used to prove that theorems are true.

Proof – logical argument backed by statements that are accepted as true.

3 types of proofs-paragraph (informal), 2 column (used most often) and flow chart Congruence – means equal. We can say that 2 lines that have the same measure are “congruent”. We use the symbol 

Ruler Postulate

The pts on any line can be paired with real numbers so that given any 2 pts P and Q on the line. P corresponds to zero and Q corresponds to a positive number.

Segment Addition Postulate

If Q is between P and R, then PQ + QR = PR If PQ + QR = PR, then Q is between P and R

Using a straight edge and compass only:

Draw a segment in your notes using a straight edge. Now, using a straight edge and compass, construct a segment congruent to the one you drew without using the markings on the ruler side of your straight edge. Explain how you did it  Using your straight edge and compass, construct a picture to explain the segment addition postulate and how it works.

N 6x - 5 30 L Find LM if L is between N and M, NL = 6x – 5, LM = 2x + 3 and NM = 30. Prove each step! Hint: Draw a picture 2x + 3 M 6x – 5 + 2x + 3 = 30 8x – 2 = 30 Segment addition postulate Substitution 8x = 32 Addition property x = 4 LM = 2(4) + 3 = 11 Division property Substitution Substitution

*Midpoint – point of a segment that divides the segment into 2 equal parts.

*Segment bisector – is a point, ray, line, line segment or plane that intersects the segment at its midpoint.

4x-1 3x+3 V M VM = MW 4x-1 = 3x + 3 x – 1 = 3 x = 4 Definition of midpoint Substitution Subtraction property Addition property Did we answer the question?

VM = 4x – 1 VM = 4(4) – 1 VM = 15 units Given Substitution Substitution W

To find the midpoint on coordinate plane Use the midpt formula:

m



x

1 

x

2 , 2

y

1 

y

2 2

m

Find the midpt of between (-3,-4) and (5,7)

x

  3 2 5

m y

  4 2  7 M = (1,1.5)

To find the coordinates of end pt given midpt.

Use the midpt formula, but solve for a different variable.

Find Q given RQ if P(4,-1) and R(3,-2).

m

8 4

x

  

x

1 

x

2 3

m

3  2

x

2  2

x

2 1 2

y

   

y

1 2  2  2  2

y

2

y

2 

y

2 (5,0)

Find Q given NQ if L(4,-6) and N(8,-9).

8 4   8  2

x

2 6 8 

x

2  12  9   2

y

2  9 

y

2 (0,-3) 2x + 11 = 4x - 5 16 = 2x 8 = x If y is midpt of xz, xy = 2x+11 and yz=4x-5, find xz xy = 2(8) + 11 xz = 2(27) = 27 = 54

d



Distance Formula



x

2 

x

1

y

2 

y

1  2 X coordinate from pt # 1 X coordinate from pt # 2 Y coordinate from pt # 1 Y coordinate from pt # 2 Commit this to memory…You are going to need it 

Find JK if J(9,-5) and K(-6,12) Distance formula:

d

   6  9   12   5  2

d

     2

d

 225  289

d

 514

d

 22 .

6

units

Pg 12 13 – 26, 28 Pg 19 3-5, 14-16 (show the properties), 17-19, 24-27, 31-33

(31 problems total for 2 days…doable  )