1. Geometry Unit 3rd Prep
Ali Adel

2. Geometry Lessons Lesson 1 Exercise Distance Between Two Points5
Lesson 1
Exercise
Distance Between Two Points
General Exercise 1 4
Lesson 2
Lesson 4
The Equation Of Straight Line
Midpoint Of Line Segment
Lesson 3
The Slope OF Straight Line 2 3

3. Distance Between Two Points

4. Distance Between Two Points
Finding The General Formula :
Right angel Triangle y
* We Want To Measure The Distance AB From Graph ∆ ACB Is Right Angel At (<C)
By Appling Pythagoras Th. On ∆ ACB y2 y1 .
B (X2,Y2)
……………………….………….……..…..
Y2 – Y1 .
…….…………..
…….…………………….
…………………………………….………..
X2 – X1 C
A (X1,Y1)
(AB) = (AC) + (CB)
(AB) = ( X2 – X1 ) + ( Y2 – Y1 )
AB = ( X2 – X1 ) + (Y2 – Y1 )
……………………….. x 2 2 2
x x2 2 2 2
__________________ √ 2 2
General Formula

5. Distance Between Two Points
Example :
Prove That The Triangle Of Vertices A ( 1 , 4 ) , B ( -1 , -2 ) , C ( 2 , -3 ) Is Right Angle Then Find Its Surface .
-Solution-
* ( AB ) = ( -1 – 1 ) + ( -2 – 4 ) = = 40 .
A ( 1 , 4 )
* ( BC ) = ( 2 – (-1) ) + ( -3 – (-2) ) = = 10
* ( AC ) = ( 2 – 1 ) + ( -3 – -4 ) = = 50
……… Pythagorean Th.
( AC ) = ( AB ) + ( BC ) .
B ( -1 , -2 ) .
C ( 2 , -3 )
Then ∆ ABC Is Right Angle At Angel B
___
___
Area = ½ √40 √ 10 = 10 Units 2

6. Midpoint Of Line Segment

7. Midpoint Of Line Segment
Definition : B .
(6,6)
- We Have Two Points A (2,2) , B (6,6) Formed A Line Seg .
- We Draw Two Normal Lines To X-Axis , Y-axis From Seg
Midpoint To Get Its Coordinates
- We’ll Find That The Midpoint Has Coordinates C (4,4)
But We Want To Get The General Formula How ?? C
(4,4) .
…………..…….. .
…………..…….. A
(2,2)

8. ⌠ Midpoint Of Line Segment Finding The General Formula : y xB
(x2,y2)
*From The Graph ∆ACM ≡ ∆MDB (ASA Case) y2 y1 . ⌠
……………………….………….……..….. M
(x,y) y .
y1-y
M(<D) = M(<C)
M(<A) = M(<M)
AM = MB
x1-x D
………………………………. .
y-y1
…….…………..
x-x1 C A
(x1,y1)
…………………………………….………..
…………………………………….
……………………….. x
AC = MD
X2 – X = X – X1
2X = X2 – X1
x x x2
X2 – X1
………… ( 1 )
_________
X = 2

9. ( ) Midpoint Of Line Segment Finding The General Formula : y x
Similarly B
(x2,y2) y2 y1 .
MC = BD
Y2 – Y = Y – Y1
2Y = Y2 – Y1
……………………….………….……..….. M
(x,y) y .
y1-y
x1-x D
………………………………. .
y-y1
…….…………..
x-x1 C A
(x1,y1)
…………………………………….………..
…………………………………….
……………………….. x
Y2 – Y1
………… ( 2 )
_________
x x x2
Y = 2
From ( 1 ) , ( 2 ) Then
( )
________ , ________
X2 – X Y2 – Y1
The Coordinates Of Midpoint M =
General Formula

10. ( ) ( ) Midpoint Of Line Segment Example : -Solution-
ABCD Parallelogram S.T A ( 3 , 2 ) , B ( 4 , -5 ) , C ( 0 , -3 ) , D ( -1 , 4 ) Find The Coordinates Of Diagonals Intersection Point .
-Solution-
The Intersection Point Of Diagonals Of Parallelogram
Bisects Its Other .
D ( -1 , 4 ) .
M Is A Midpoint .
A ( 3 , 2 )
( )
________ , ________
X2 – X Y2 – Y1
M = M
( )
– 3
________ , ________ .
M =
C ( 0 , -3 ) .
B ( 4 , -5 )
M = ( , ) ⅔ -½

11. The Slope OF Straight Line

12. ᵩ Slope = Tan( ) = The Slope OF Straight Line
Finding The General Formula :
Right angel Triangle y
* We Want To Find The Slope Of AB From Graph ∆ ACB Is Right Angel At (<C)
From Trigonometry : B
(x2,y2) y2 y1 .
……………………….………….……..….. H
Y2 – Y1 O ᵩ .
…….…………..
…….…………………….
X2 – X1 C A
(x1,y1)
…………………………………….……….. A
Slope = Tan( ) ᵩ
……………………….. x
Opposite
________
___ O A
x x2 = =
Adjacent =
___ BC CA =
_______
Y2 – Y1
X2 – X1 ᵩ
Slope = Tan( ) =
_______
Y2 – Y1
X2 – X1
General Formula

13. The Slope OF Straight Liney
Special Cases :
Slope Of Line Parallel To X-Axis :
Line L is Parallel To X-axis
Slope Of Line L = M = Zero L x
______________________________
Slope Of Line Parallel To Y-Axis : y
Line L is Parallel To Y-axis
Slope Of Line L ( Undefined ) L x

14. The Slope OF Straight Liney
Special Cases :
Relation Between Slopes Of Two Parallel Lines :
IF L1 // L2
IF M1 , M2 Are The Slopes Of L1 , L2 Recpectly Then
L 1
L 2 x
M1 = M2
______________________________
Relation Between Slopes Of Two Perpendicular Lines :
IF L1 ⊥ L2
IF M1 , M2 Are The Slopes Of L1 , L2 Recpectly Then y
L 1
M1 x M2 = -1
L 2 x

15. The Slope OF Straight Line
Example :
Prove By Using The Slope That The Points A( -1 , 3 ) , B ( 5 , 1 ) , C ( 6 , 4 ) , D ( 0 , 6 ) Are The Vertices Of The Rectangle .
-Solution-
* Slope Of AB
D ( 0 , 6 ) .
C ( 6 , 4 )
_______
1 – 3
5 + 1
___ -1 3
m1 = = . .
A ( -1 , 3 )
* Slope Of DC .
B ( 5 , 1 )
_______
6 – 4
0 – 6
___ -1 3
m2 = =
m1 = m2
Then AB // DC …….. ( 1 )

16. The Slope OF Straight Line
Example :
Prove By Using The Slope That The Points A( -1 , 3 ) , B ( 5 , 1 ) , C ( 6 , 4 ) , D ( 0 , 6 ) Are The Vertices Of The Rectangle .
-Solution-
* Slope Of AD
D ( 0 , 6 ) .
C ( 6 , 4 )
_______
6 – 3
0 + 1
m3 = = 3 . .
A ( -1 , 3 )
* Slope Of BC .
B ( 5 , 1 )
_______
4 – 1
6 – 5
m4 = = 3
m3 = m4
Then AD // BC …….. ( 2 )

17. The Slope OF Straight Line
Example :
Prove By Using The Slope That The Points A( -1 , 3 ) , B ( 5 , 1 ) , C ( 6 , 4 ) , D ( 0 , 6 ) Are The Vertices Of The Rectangle .
-Solution-
From ( 1 ) , ( 2 ) Then :
D ( 0 , 6 ) .
C ( 6 , 4 )
ABCD Is Parallelogram
……… ( 3 ) . .
We Notice That :
A ( -1 , 3 ) .
m1 x m3 = -1
B ( 5 , 1 )
Then
AB ⊥ AD
……… ( 4 )
From ( 3 ) , ( 4 ) Then ABCD Is Rectangle

18. The Equation Of Straight Line

19. The Equation Of Straight Line
Discussion :
In This Lesson We’ll Know How To Get The Equation Of Straight Line Given its slope and its Y-Intercept so we should know First the Different forms of Straight line equation
Graphs Determines What’s The Meaning Of Y-Intercept ( C ) y y L C L C x x

20. The Equation Of Straight Line
The Forms Of Straight Line Equation
First Form :
Slope y
Y = mX+ b
Second Form : L
Y-Intercept C x
aX + bY + C = 0
X Coefficient
Y Coefficient
In This Case m =
- X Coefficient -a b
______________
___ =
( a,b ≠ 0 )
Y Coefficient

21. The Equation Of Straight Line
Example :
Sketch The Straight Line 3X + 4Y – 12 = 0 Then Find Its Slope
-Solution-
* Put Y = 0 To Get Point Of Intersection With X-axis
A ( 4 , 0 )
B ( 0 , 3 )
* Put X = 0 To Get Point Of Intersection With Y-axis .
B ( 0 , 3 )
A ( 4 , 0 ) .
* We Can Sketch The Line & We Can Get Slope From Eq
By Comparing Its Coefficients With General Form Of
Straight Line Eq.
3X + 4Y – 12 = 0
aX + bY – 12 = 0 -a b
___
= -¾
Slope =

22. General Exercise

23. General Exercise Example 1 : Solve It By Yourself
State The Kind Of Triangle Whose Vertices are The Points A ( -2 , 4 ) , B ( 3 , -1 ) , C ( 4 , 5 ) With
Respect To Its Sides
Solve It By Yourself

24. General Exercise Example 2 : Solve It By Yourself
If C ( 6 , -4 ) Is The Midpoint Of AB Find Coordinates Of B If A ( 5 , -3 )
Solve It By Yourself

25. General Exercise Example 3 : Solve It By Yourself
If A ( -3 , 4 ) , C ( -3 , -2 ) , B ( 1 , 2 ) , D ( -3 , 2 ) Prove That AC ⊥ BD
Solve It By Yourself

26. General Exercise Example 4 : Solve It By Yourself
If A ( -3 , 4 ) , B ( 5 , -1 ) , C ( 3 , 5 ) Find The Equation Of Straight Line Passing Through The Vertex
( A ) & Bisecting BC
Solve It By Yourself

27. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
A Circle Its Center Is The Origin Point & Radius Length 2 Units . Which Of
The Following Points Belongs To The Circle .
( 1 , 2 )
( -2 , 1)
___
( √3 , 1 )
( 1 , 2 )

28. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
A Circle Its Center Is The Origin Point & Radius Length 2 Units . Which Of
The Following Points Belongs To The Circle .
( 1 , 2 )
( -2 , 1)
___
( √3 , 1 )
( 1 , 2 )
Click Here To Continue

29. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
A Circle Its Center Is The Origin Point & Radius Length 2 Units . Which Of
The Following Points Belongs To The Circle .
( 1 , 2 )
( -2 , 1)
___
( √3 , 1 )
( 1 , 2 )
Click Here To Try Again

30. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(B) If The Point Of The Origin Is The Midpoint Of Straight Segment AB , Where
A ( 5 , -2 ) Then The Coordinates Of The Point B Are
( -5 , 2 )
( -5 , 1)
( 4 , 2 )
( 3 , -1 )

31. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(B) If The Point Of The Origin Is The Midpoint Of Straight Segment AB , Where
A ( 5 , -2 ) Then The Coordinates Of The Point B Are
( -5 , 2 )
( -5 , 1)
( 4 , 2 )
( 3 , -1 )
Click Here To Continue

32. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(B) If The Point Of The Origin Is The Midpoint Of Straight Segment AB , Where
A ( 5 , -2 ) Then The Coordinates Of The Point B Are
( -5 , 2 )
( -5 , 1)
( 4 , 2 )
( 3 , -1 )
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33. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Passes Through To Points A ( 3 , 1 ) , B ( 8 , 11 ) Is
Equal To
-11 3 -1 2

34. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Passes Through To Points A ( 3 , 1 ) , B ( 8 , 11 ) Is
Equal To
-11 3 -1 2
Click Here To Continue

35. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Passes Through To Points A ( 3 , 1 ) , B ( 8 , 11 ) Is
Equal To
-11 3 -1 2
Click Here To Try Again

36. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Whose Equation Is Y = 3 X + 9 9 6 3 1

37. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Whose Equation Is Y = 3 X + 9 9 6 3 1

38. Choose The Correct Answer :
General Exercise
Choose The Correct Answer :
(C) The Slope Of Straight Line Whose Equation Is Y = 3 X + 9 9 6 3 1
Click Here To Try Again