Geometry Unit 3rd Prep Ali Adel

Geometry Lessons Lesson 1 Exercise Distance Between Two Points 5 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

Distance Between Two Points

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 x1 x2 2 2 2 __________________ √ 2 2 General Formula

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- 2 2 2 * ( AB ) = ( -1 – 1 ) + ( -2 – 4 ) = 4 + 36 = 40 . A ( 1 , 4 ) * ( BC ) = ( 2 – (-1) ) + ( -3 – (-2) ) = 9 + 1 = 10 2 2 2 * ( AC ) = ( 2 – 1 ) + ( -3 – -4 ) = 1 + 49 = 50 2 2 2 ……… Pythagorean Th. ( AC ) = ( AB ) + ( BC ) 2 2 2 . B ( -1 , -2 ) . C ( 2 , -3 ) Then ∆ ABC Is Right Angle At Angel B ___ ___ Area = ½ √40 √ 10 = 10 Units 2

Midpoint Of Line Segment

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)

⌠ Midpoint Of Line Segment Finding The General Formula : y x B (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 x1 x x2 X2 – X1 ………… ( 1 ) _________ X = 2

( ) 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 ) _________ x1 x x2 Y = 2 From ( 1 ) , ( 2 ) Then ( ) ________ , ________ X2 – X1 Y2 – Y1 The Coordinates Of Midpoint M = 2 2 General Formula

( ) ( ) 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 – X1 Y2 – Y1 M = 2 2 M ( ) 3 + 0 2 – 3 ________ , ________ . M = C ( 0 , -3 ) 2 2 . B ( 4 , -5 ) M = ( , ) ⅔ -½

The Slope OF Straight Line

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 x1 x2 = = Adjacent = ___ BC CA = _______ Y2 – Y1 X2 – X1 ᵩ Slope = Tan( ) = _______ Y2 – Y1 X2 – X1 General Formula

The Slope OF Straight Line y 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

The Slope OF Straight Line y 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

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 )

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 )

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

The Equation Of Straight Line

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

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

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 =

General Exercise

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

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

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

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

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 )

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

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

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 )

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

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 Try Again

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

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

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

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

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

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