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PROBLEM 2A

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  1. STANDARDS 13, 17 PLANES LINES SEGMENT ADDITION POSTULATE PROBLEM 1B PROBLEM 1A MIDPOINT PROBLEM 2A PROBLEM 2B COORDINATE GEOMETRY PROBLEM 3A PROBLEM 3B DISTANCE FORMULA PROBLEM 6 PROBLEM 4 PROBLEM 5 PYTHAGOREAN THEOREM PROBLEM 7 PROBLEM 8 MIDPOINT FORMULA PROBLEM 9A PROBLEM 9B PROBLEM 10A PROBLEM 10B END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  2. STANDARD 13: Students prove relationships between angles in polygons using properties of complementary, supplementary, vertical and exterior angles. STANDARD 15: Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles. STANDARD 17: Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  3. ESTÁNDAR 13: Los estudiantes prueban relaciones entre ángulos en polígonos usando propiedades de ángulos complementarios, suplementarios, verticales y ángulos exteriores. ESTÁNDAR 15: Los estudiantes usan el Teorema de Pitágoras para determinar distancias y encontrar las longitudes de los lados de triángulos. ESTÁNDAR 17: Los estudiantes prueban teoremas usando geometría coordenada, incluyendo el punto medio de un segmento, la fórmula de la distancia y varias formas de ecuaciones de líneas y círculos. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  4. Line AB H Line HI M N I Line MN LINES STANDARD 13 B p A or Line p How do we call this line? is VERTICAL What about this other? is HORIZONTAL PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  5. Line CF Line CD Line FC Line EC Line CE Line DE Line DF Line EF Line DC LINES STANDARD 13 F m E D C How many different ways can we call Line m? OR Can you figure out other names? IN GENERAL A LINE IS NAMED BY TWO POINTS. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  6. If we have line DE D E STANDARD 13 SEGMENTS and we take one part of the line PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  7. If we have line DE LINE SEGMENT DE S T U D E TU TS ST UT SU US STANDARD 13 SEGMENTS and we take one part of the line E D then this part is called: Can you name the different line segments in the following line: SEGMENTS: OR PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  8. A B B A E D M N AB STANDARD 13 RAYS This is RAY What are the differences between a Line, a Line Segment and a Ray? The line is infinite and never ends at either side. The line segment has two endpoints. The ray has on one side one endpoint at the other side it never ends it goes on to infinite. What do they have in common? They are named using TWO POINTS. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  9. F E D A C B D C Are points A, B, C, and D collinear? STANDARD 13 COLLINEAR VS NONCOLLINEAR Points C, D, E and F: are they COLLINEAR? Yes, they are COLLINEAR because they lie in the same line No, they are NONCOLLINEAR because they don’t lie in the same line. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  10. l B C E A D m STANDARD 13 Can you explain where the following two lines intersect? They INTERSECT at point E. NO Can they intersect at other point? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  11. A C D B STANDARD 13 PLANES M This is: PLANE M PLANE ADC PLANE BDC PLANE ABD In general a Plane can be named using three non-collinear points. The figure above represents a PLANE, which is a flat surface that has no end at any of the sides. What examples can you give of objects lying in a PLANE? • The wall of a house • The lid of a shoebox. • The ground of a football field. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  12. A n P m B M l Planes M and Pintersect at line AB. STANDARD 13 Can you describe the INTERSECTION of planes M and P? PLANES ALWAYS INTERSECT AT A LINE. They intersect at point B. LINES ALWAYS INTERSECT AT A POINT. Where do lines m and l intersect? At point A. Where do lines n and l intersect? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  13. A K L M H P B Q C STANDARD 13 D Are Points L, K, and MCOPLANAR? Yes, they are COPLANARbecause they LIE ON THE SAME PLANE P. Is point H, coplanar with points L, K, and M? No, because it lies on plane Q and points L, K, and M are in different plane, on plane P. NON-COPLANAR points are points that lie in different planes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  14. A P B Q STANDARD 13 D C On what planes does point C lie? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  15. STANDARD 13 A D B C P Q On what planes does point C lies? On planes P andQ. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  16. STANDARD 13 A P D B Q C On what planes does point D lie? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  17. A P D B Q C STANDARD 13 On what planes does point D lie? It only lies on plane Q. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  18. A P D B Q C STANDARD 13 k On what plane is line k lying? Since points B and D lie on plane Q then line k lies on its entirety on plane Q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  19. STANDARD 13 SUMMARIZING FINDINGS: • Through any two points there is exactly one line. • Through any three points not on the same line there is exactly one plane or through any three points non-collinear there is one plane. • A line contains at least two points. • A plane contains at least three points not on the same line. • A plane contains at least three non-collinear points. • If two points lie in a plane, the entire line containing those points lies in that plane. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  20. A C B AB AC BC SEGMENT ADDITION POSTULATE. • If B is between A and C then AB + BC = AC. • If AB + BC = AC, then B is between A and C. STANDARD 17 + = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  21. A C B AB AC BC Now finding AB and BC: + = AB AC BC 10 10 STANDARD 17 Find the length for AB and BC if AC = 60 and AB = 4x + 6 and BC= 6x + 14. (B is between A and C) Applying Segment Addition Postulate: + = 4x +6 + 6x + 14 = 60 AB = 4x + 6 BC = 6x + 14 4x + 6x + 6 + 14 = 60 4 = 4( ) + 6 = 6( ) + 14 4 = 16 + 6 10x + 20 = 60 = 24 + 14 -20 -20 = 22 = 38 10x = 40 Verifying the solution: x = 4 22 + 38 = 60 60 = 60 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  22. E G F EF EG FG Now finding EF and FG: + = EF EG FG 10 10 STANDARD 17 Find the length of EF and FG if EG = 80 and EF = 3x + 8 and FG= 7x + 12. (F is between E and G) Applying Segment Addition Postulate: + = 3x +8 + 7x + 12 = 80 EF = 3x + 8 FG = 7x + 12 3x + 7x + 8 + 12 = 80 6 = 3( ) + 8 = 7( ) + 12 6 = 18 + 8 10x + 20 = 80 = 42 + 12 -20 -20 = 26 = 54 10x = 60 Verifying the solution: x = 6 26 + 54 = 80 80 = 80 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  23. and we place point B at the same distance from point A than from Point C, then: A C B AB BC If we have segment AC, Means congruent and then point B is THE MIDPOINT OF SEGMENT AC. Point B is also BISECTING segment AC, because it is dividing it into two halves. STANDARD 17 MIDPOINT OF A SEGMENT: AB = BC PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  24. AC is bisected by ED. AB= 6X + 8 and BC=4X + 18. Find the length for AC. A D B C AB BC E 2 2 AC = AB + BC STANDARD 17 Applying Segment Addition Postulate: Now finding AB: AB = BC AB = 6X + 8 6X + 8 = 4X + 18 AC = AB + BC = 6( ) + 8 5 -8 -8 AC = 38 + 38 = 30 + 8 6X = 4X + 10 AC = 76 = 38 -4X -4X Since AB =BC 2X = 10 BC = 38 X = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  25. RT is bisected by VU. RS= 8X + 4 and ST=4X + 28. Find the length for RT. R V S T RS ST U 4 4 RT = RS + ST STANDARD 17 Applying Segment Addition Postulate: Now finding RS: RS = ST RS = 8X + 4 8X + 4 = 4X + 28 RT = RS + ST = 8( ) + 4 6 -4 -4 RT = 52 + 52 = 48 + 4 8X = 4X + 24 RT = 104 = 52 -4X -4X Since RS =ST 4X = 24 ST = 52 X = 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  26. Origin STANDARD 17 CARTESIAN COORDINATE PLANE y-axis 10 Quadrant I Quadrant II 8 6 4 2 O -10 x-axis -8 -6 -4 -2 2 8 6 10 4 -2 -4 Quadrant III Quadrant IV -6 -8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  27. y-coordinate x-coordinate STANDARD 17 CARTESIAN COORDINATE PLANE y-axis 10 8 6 (9, 4) 4 2 O -10 x-axis -8 -6 -4 -2 2 8 6 10 4 -2 -4 -6 -8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  28. y-coordinate x-coordinate STANDARD 17 CARTESIAN COORDINATE PLANE y-axis 10 8 6 4 2 O -10 x-axis -8 -6 -4 -2 2 8 6 10 4 -2 -4 -6 (10,-8) -8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  29. y-coordinate x-coordinate STANDARD 17 CARTESIAN COORDINATE PLANE y-axis 10 8 6 4 2 O -10 x-axis -8 -6 -4 -2 2 8 6 10 4 -2 (-9,-3) -4 -6 -8 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  30. -6 2 -4 0 4 8 6 -2 10 12 E D H Find measure of DE, and EH: STANDARD 17 DISTANCE FORMULA in a number line is given by: |a – b| EH = |12 – (-2)| DE = |-2 – (-6)| = |12 + 2| = |-2 + 6| = |14| = |4| = 14 = 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  31. -6 2 -4 0 4 8 6 -2 10 12 Q R T Find measure of RT, and QT: STANDARD 17 DISTANCE FORMULA in a number line is given by: |a – b| RT = |12 – (-6)| QT = |12 – 6| = |12 + 6| = |6| = 6 = |18| = 18 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  32. y 10 8 6 4 2 -10 x -8 -6 -4 -2 2 8 6 10 4 -2 -4 -6 -8 2 2 d = (x –x ) + (y –y ) 2 2 1 1 x x 2 1 y y 1 2 STANDARD 17 Distance Formula between two points in a plane: , , PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  33. Remember: B 2 2 2 2 AB= ( - ) + ( - ) AB= ( -4 ) + ( -3 ) y A 9 8 = 16 + 9 7 6 = 25 5 4 3 2 2 2 1 d = (x –x ) + (y –y ) 2 2 1 1 1 6 4 9 3 5 8 10 2 7 x x x 1 2 y y 2 1 STANDARD 17 Find the distance between points at A(2, 1) and B(6,4). 2 1 6 4 AB=5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  34. y 10 2 2 2 2 2 2 = ( 5 ) + ( -1 ) d = ( - ) + ( - ) = ( + ) + ( + ) 8 6 4 d= 26 2 -10 x -8 -6 -4 -2 2 8 6 10 4 -2 -4 2 -6 5 3 -6 -8 = 25 + 1 2 2 d = (x –x ) + (y –y ) 2 2 1 1 x x 1 2 y y 2 1 STANDARD 17 Find the distance between (-3,-5) and (2,-6). =(-3,-5) =(2,-6) 2 -6 -5 -3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  35. 2 2 10 = ( - ) + ( - ) 2 2 2 2 a -6 2 10 10 = (-6-a) + (-8) (-1) (-1) 2 2 d = (x –x ) + (y –y ) 2 36 = (-6-a) 2 2 1 1 2 100 = (-6-a) + 64 x x 1 2 y y 1 2 STANDARD 17 Find the value of a, so that the distance between (-6,2) and (a,10) be 10 units. We use the distance formula: Solving this absolute value equation: 6 = |-6-a| 6 = -6-a 6 = -(-6-a) +6 +6 6 = 6 + a 12 = -a -6 -6 a = -12 a = 0 -64 -64 Check: 6 = |-6-a| -12 6 =|-6- ( )| 6 =|-6- ( )| 0 6 = |-6-a| 6 =|-6+12| 6 = |-6| 6 =|6| 6 = 6 6 = 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  36. hypotenuse LEG Right Angle = 90° LEG STANDARD 15 Right triangle parts: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  37. y + z = x 2 2 2 STANDARD 15 Pythagorean Theorem: x y z The square of the hypotenuse is equal to the sum of the square of the legs. The Pythagorean Theorem applies ONLY to RIGHT TRIANGLES! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  38. y + z = x 2 2 2 2 2 4 + 3 = x 2 16 + 9 = x 2 2 25 = x 2 25 = x STANDARD 15 Find the value for x: x= ? y= 4 z= 3 |x|= 5 x= -5 x= 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  39. y + z = x 2 2 2 2 2 2 8 + z = 10 64 + z = 100 2 2 2 z = 36 z = 36 STANDARD 15 Find the value for z: y= 8 z= ? x= 10 -64 -64 |z|= 6 z= -6 z= 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  40. Find Midpoint Q of RT: a + b -6 2 -4 0 4 8 6 -2 10 12 Q 2 R T -6 + 12 = 6 2 2 STANDARD 17 MIDPOINT FORMULA in a number line is given by: = 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  41. Find Midpoint R of KL: a + b -4 4 -2 2 6 10 8 0 12 14 R 2 K L -4 + 14 = 10 2 2 STANDARD 17 MIDPOINT FORMULA in a number line is given by: = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  42. If a line segment has endpoints at and , then the midpoint of the line segment has coordinates: + + , 2 2 y , 5 4 3 (x,y) 2 1 -5 x -4 -3 -2 2 = 1 4 -1 3 5 -1 , -2 -3 x x x x x x 1 1 2 1 2 2 y y y y y y -4 1 1 2 2 1 2 STANDARD 17 Midpoint of a Line Segment: y x, PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  43. + + , 2 2 y 9 1 9 8 6 8 (9,8) 14 10 (5,7) , 7 2 2 6 (1,6) + y x, , 5 2 4 = = = = + y x, 7 5, 3 2 2 x x x x 1 2 1 2 1 y y y y 1 2 1 2 1 6 4 9 3 5 8 10 2 7 x STANDARD 17 y Using: x, Find the midpoint of the line segment that connects points (1,6) and (9,8). Show it graphically. y x, PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  44. + + , 2 2 y 9 4 8 9 7 8 (8,9) 16 12 (6,8) , 7 2 2 6 (4,7) + y x, , 5 2 4 = = = = + y x, 8 6, 3 2 2 x x x x 1 2 1 2 1 y y y y 1 2 1 2 1 6 4 9 3 5 8 10 2 7 x STANDARD 17 y Using: x, Find the midpoint of the line segment that connects points (4,7) and (8,9). Show it graphically. y x, PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  45. y 20 Given the coordinates of one endpoint of KL are K(-2,-6) and its midpoint M(8, 5). What are the coordinates of the other endpoint L. Graph them. 16 12 y x, 8 4 L y x, -20 x -16 -12 -8 -4 16 12 20 8 4 -4 M -8 + + , -2 -12 -6 5 8, 2 2 -16 K , 16 18, -2 -6 8= 5= 16 =-2 + 10 =-6 + + + , 2 2 =16 =18 = = = + + 2 2 x x x x x x x x 2 2 2 1 2 2 1 2 y y y y y y y y 1 2 2 2 2 1 2 2 STANDARD 17 Using the Midpoint Formula: (2) (2) (2) (2) +6 +6 +2 +2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

  46. y 20 Given the coordinates of one endpoint of KL are K(-1,-5) and its midpoint M(7, 4). What are the coordinates of the other endpoint L. Graph them. 16 12 y x, 8 4 L y x, -20 x -16 -12 -8 -4 16 12 20 8 4 -4 M -8 + + , -1 -12 -5 4 7, 2 2 -16 K , 13 15, -1 -5 7= 4= 14 =-1 + 8 =-5 + + + , 2 2 =13 =15 = = = + + 2 2 x x x x x x x x 2 2 2 1 2 2 1 2 y y y y y y y y 1 2 2 2 2 1 2 2 STANDARD 17 Using the Midpoint Formula: (2) (2) (2) (2) +5 +5 +1 +1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved