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Unit 6 – Connecting Algebra and Geometry

Unit 6 – Connecting Algebra and Geometry. Writing Linear Equations Given Slope and y-intercept. y = mx + b Given Slope and a point. Given 2 points Parallel and Perpendicular Lines Use coordinates to prove simple geometric theorems algebraically. Triangles Parallelograms Rectangles

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Unit 6 – Connecting Algebra and Geometry

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  1. Unit 6 – Connecting Algebra and Geometry • Writing Linear Equations • Given Slope and y-intercept. y = mx + b • Given Slope and a point. • Given 2 points • Parallel and Perpendicular Lines • Use coordinates to prove simple geometric theorems algebraically. • Triangles • Parallelograms • Rectangles • Circle centers and Radii • Ratios and Proportions • Partitions of a Line Segment

  2. I. How to Write an Equation of a Line Given m and b 1. Write down y = mx + b 2. Substituteslope for m and y-intercept for b. 3. Simplify the equation

  3. Write the equation of the line given m and b. Ex. 1 Slope is -5 and y-intercept is 2 y = -5x + 2 Ex. 2 Slope is -1/2 and y-intercept is -2 y = -½x – 2

  4. Write the equation of the line given m and b. Ex. 3 Slope is 0 and y-intercept is 3 y = 3 Ex. 4 Slope is 1/3 and y-intercept is 0

  5. 11. & 12. Write the equation of each graph     y = 2x – 3 y = -3/2x + 3

  6. I. How to Write an Equation of a LineGiven m and a point (x,y) 1. Write down y = mx + b 2. Substituteslope for m and pointfor ‘x’ and ‘y’. 3. Solve for b and substitute.

  7. Write the equation of the line given m and point (x,y). Ex. 1 Slope is -5 and goes through (1,-3) y = -5x + 2 Ex. 2 Slope is -1/2 and goes through (4,-4) y = -½x – 2

  8. I. How to Write an Equation of a LineGiven 2 points (x1,y1) and (x2,y2) 1. Write down y = mx + b Findm using m Substitue for x and ywith one of the 2 points given. Solve for b.

  9. Write the equation of the line given the following 2 points. Ex. 1(-2,-1) and (-2,3) x = -2 Ex. 2(-2,6) and (2,8) y = ½x + 7

  10. How to Write an Equation of a Line PARALLEL to another and given a point • 1. Given equation should be solved for y (y = mx + b) • Write down the slope of that line • Substitute m and (x, y) in y = mx + b. • Solve for b. • Write the equation using m and b.

  11. Write the equation of the line parallel to another and given a point. Ex. 1Write a line parallel to the line 3x + 3y = – 7 and passes through the point (-4, -4). Y = -x-8 Ex. 2 Write a line parallel to the line y = – 7 and passes through the point (-4, -4). y = -4

  12. Write the equation of the line parallel to another and given a point. Ex. 3Write a line parallel to the line -x + 3y = 18 and passes through the point (3, 5). Y = 1/3x + 4 Ex. 4 Write a line parallel to the line with an undefined slope and passes through the point (3, 5). X = 3

  13. Write the equation of the line parallel to another and given a point. Ex. 3Write a line parallel to the line Below and passes through (3, 5). y = 2x -1

  14. How to Write an Equation of a Line Perpendicular to another and given a point • 1. Given equation should be solved for y (y = mx + b) • Write down the slope of that line and compute its negative reciprocal • Substitute m and (x, y) in y = mx + b. • Solve for b. • Write the equation using m and b.

  15. Write the equation of the line parallel to another and given a point. Ex. 1Write a line perpendicularto the line 4x + 3y = – 7 and passes through the point (-4, -4). y = 3/4x - 1 Ex. 2 Write a line pependicularto the line y = – 7 and passes through the point (-3, -4). x = -3

  16. Write the equation of the line parallel to another and given a point. Ex. 3Write a line perpendicularto the line -x + 3y = 18 and passes through the point (3, 5). Y = -3x + 14 Ex. 4 Write a line perpendicularto the line with an undefined slope and passes through the point (3, 5). y = 5

  17. Write the equation of the line parallel to another and given a point. Ex. 3Write a line perpendicularto the line below and passes through (3, 5). y = 2/3x +1

  18. Using parallel and perpendicular lines to prove properties of polygons.. Two lines that intersect at a right angle (90°) are perpendicular. 2 lines are perpendicular if and only if their slopes are negative reciprocals.

  19. Determine if the triangle is a right triangle by applying slopes to show that is has a right angle The slope ofOPis 2 – 0 3 – 0 1 . – 2. The slope ofOQis = = – 2 – 0 2 6 – 0 so OPOQand POQ is a right angle. Using parallel and perpendicular lines to prove properties of polygons..

  20. Using parallel and perpendicular lines to prove properties of polygons.. Determine if the quadrilateral is a parallelogram and a rectangle by using its slopes. mDA= 3/7 mCB= 3/7 mAB= -2 mDC= -2 Parallelogram? Rectangle?

  21. Distance Formula

  22. Use the distance formula to find the distance between the following points. (1, 4) and (-2, 3) (10, 5) and (40, 45). D = 3.16 D = 50

  23. Use the distance formula to find area and perimeter of polygons. • Find the perimeter and area of the following figures. Rectangle P = 24.2 A = 34.2 Triangle P = 26.2 A = 32

  24. PYTHAGOREAN THEOREM Use the Pythagorean theorem to find the missing length of the right triangle. Round to the nearest hundredth. 1. 2. c 2 17 8 5 b b = 15 c = 5.4

  25. Determine if the triangle is a right triangle by applying the distance formula and Pythagorean Theorem 2 2 ( – ) 6 (– 1 ) ) 3 – ( 2 7.1 + = = 2 2 ( – ( ) (– 1 ) ) 0 2 – 0 2.2 + = 5 50 2 2 ( – ( ) 6 ) 0 – 0 3 6.7 + = 45 Use the distance formula to prove properties of triangles. OP= OQ = PQ =

  26. Using distance formula to prove properties of polygons.. Determine if the parallelogram is a rectangle by using its side lengths. AB= BC= AD= Opposite sides are congruent = Rectangle

  27. Midpoint Formula • Used to find the center of a line segment

  28. Use the midpoint formula to find the center of the following points. • Find the midpoint between: • A) (2, 7) and (14, 9) • B) (-5, 8) and (2, - 4)

  29. Find Endpoints given midpoint and other endpoint (x1 , y1) • Used to find the center of a line segment Plug in midpoint. Substitute given endpoint for x1 and y1 Set ‘x’ part of formula equal to x-value of midpoint and ‘y’ part of formula equal to y-value of midpoint. Solve for x1and y1

  30. Use the midpoint formula to find the other endpoint of a line segment • Find the coordinates of the other endpoint of a segment with an endpoint of (2, 7) and a midpoint (5, 10). • Find the coordinates of the other endpoint of a segment with an endpoint of (-3, 4) and a midpoint (1, -2). • Find the coordinates of the other endpoint of a segment with an endpoint of (4, -3) and a midpoint (-2, 3). (-1 , 13) (-5 , 10) (-10 , -9)

  31. Using midpoint formula to prove properties of polygons.. Determine if the parallelogram is a rectangle by showing its diagonals bisect each other. Identify the point of bisection as the point M. Use the distance formula to prove that M is the midpoint of DB by showing DM = MB mdptDB= (-.5,1) = MdptAC Therefore M(-.5,1)

  32. Properties of Circles and Radii Circumference Area of a circle 2 p r = C A = p r2 or dp = C

  33. Use the formulas of circumference and area to evaluate the properties of a circle • Find the circumference of a circle with a diameter of 11ft. • Find he area of a circle with a circumference of 16π m • Find the diameter of a circle with a circumference of 27ft. • Find the area of a circle with a diameter of 18 cm. • Find the circumference of a circle with area of 121 π cm2 • Find the diameter of a circle with area of 50 cm2 11pi = 34.56 ft 64pi = 201.06 64pi = 201.06 cm2 64pi = 201.06 cm2 D = 8.59 ft 81pi = 254.47 cm2 22pi = 69.12 cm D = 3.98 cm

  34. Using distance formula to prove properties of a circle. • Determine whether Point A lies on the circle whose center is Point C and which contains the Point P. • Find radius ( distance from Center to Point on circle) • Check if distance from A to C is equal to radius

  35. Using distance formula to prove properties of a circle. • Determine whether Point A lies on the circle whose center is Point C and which contains the Point P. • Point A(3, √7); Point C(0, 0); Point P(0, 4) • Point A(-1, 2 + √21); Point C(1, 2); Point P(1, 5) • Point A(3, 4 + √7); Point C(4, 3); Point P(4, 7) Yes Yes No

  36. Ratios and Partitions • Partition a directed segment into a given ratio h:k given a line segment with endpoints (x1, y1) and (x2, y2)

  37. Ratios and Partitions • Find a point P on the segment AB that partitions it to a 1:2 ratio. • Find a point Q on segment BC that breaks it into a 3:2 ratio. C A B

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