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Geometry Chapter 13 Review: Distance, Circle Equations & Slopes

Explore distance between points, equations of circles, calculating slopes, and solving systems of equations in a coordinate plane. Practice exercises included.

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Geometry Chapter 13 Review: Distance, Circle Equations & Slopes

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  1. Geometry Chapter 13 Review

  2. The distance d between points and is: Why? Let’s try an example to find out! (-3, 4). Example 2 Find the distance between (–3, 4) and (1, –4). 4 . (1, -4) 4√5 8 Pythagorean Theorem!

  3. An equation of the circle with center (a, b) and radius r is: How could this be a circle? Let’s analyze (x – 0)2 + (y – 0)2 = 81 to see if it really is a circle!!

  4. Find the center and radius of each circle. Sketch the graph. 4. 5. Center: (2, -4) Radius = 3 .

  5. Example 1b: Find the slope of the line. y y2 – y1 slope = x2 – x1 -5– (-2) = 3 – (- 1) . x (-1, -2) - 3 = 4 . (3, -5) 3 __ - The slope of the line is 4

  6. Positive Slope Positive Slope Slope = 0 Greater than 1 Less than 1 Uphill Uphill Steep Flatter Negative Slope Undefined Slope Negative Slope Less than 1 Greater than 1 Running up the hill is undefined! Downhill Downhill Flatter Steep

  7. A line with slope 4/3 passes through points (4, -5) and (-2, __ ). y -13 Use the slope formula to find the missing y coordinate. y – (-5) 4 = Simplify and solve as a proportion 3 -2 – 4 y + 5 -24 = 3y + 15 4 = -6 -39 = 3y 3 y = -13

  8. Parallel lines have slopes that are equal. • Perpendicular lines have slopes that are opposite inverses(change the sign and flip).

  9. The Midpoint Formula The midpoint of the segment that joins points (x1,y1) and (x2,y2) is the point • (6,8) • (1,5) • (-4,2)

  10. Exercises 3. M (3,5) A (0,1) B (x,y) This is the midpoint To find the coordinates of B: x-coordinate: y-coordinate: 0 + x 1 + y 3 = 5 = 2 2 6 = 0 + x 10 = 1 + y (6,9) x = 6 y = 9

  11. x y Try the cover up method!!! 0 2 3 0 .(0, 2) . (3, 0)

  12. . . . .(0, 4) . . . . . (-6, 2) (-1, 2) (6, 2) yorizontal Why? Thus y=2!!

  13. -3x -3x -4y = -3x + 10 -4 -4 -4 y = 3/4x – 5/2 -5/2 3/4

  14. IV. Systems of Equations: Two lines in a coordinate plane can do two things: (1) intersect (perpendicular or not) (2) not intersect (parallel) .(2,4) 2x + (2x) = 8 ( ) 4x = 8 ( ) x = 2 y = -2x + 8 Substitute 2 back in for x in the easier equation!! Isolate a variable first. This is already done. Then substitute. y = 2x y = 2x Graph 2x + y = 8 y = 2(2) -2x -2x y = -2x + 8 y = 4 Graph y = 2x The solution to the system is (2, 4)

  15. IV. Systems of Equations: Two lines in a coordinate plane can do two things: (1) intersect (perpendicular or not) (2) not intersect (parallel) 4(2) + 2y = 12 . (2,2) 4x + 2y = 12 ( )2 8 + 2y = 12 -8 -8 2y = 4 y = 3/2x – 1 7x = 14 y = -2x + 6 y = 2 x = 2 Substitute 2 back in for x in the easier equation!! Graph 2x + y = 6 The solution to the system is (2, 2) -2x -2x y = -2x + 6 Graph 3x – 2y = 2 -3x -3x -2y = -3x + 2 -2 -2 -2 y = 3/2x – 1

  16. Given x and y intercepts: 1. x-int: 2 y-int: -3 (2,0) (0,-3) Notice that the slope is rise 3 (-3) ● - or (2,0) run 2 2 ● y-int (0,-3) or opposite x-int. The y intercept (b) of -3 is given 3 The equation in slope intercept form is y = x - 3 2

  17. Given Intercepts To write the equation in slope-intercept form use the pattern : y-intercept + y-intercept y = x x-intercept b slope m

  18. Part IV #1: Given 2 points. (1,2) and (4,7) Step 1: Compute slope You can check with other point: 7 = 5/3(4) + 1/3 Step 2: Use PS Form 7 = 20/3 + 1/3 Using (1, 2) 7 = 21/3 check! 7 = 7 Step 3: Simplify to SI Form +2 y = 5/3x + 1/3

  19. Part VI #5: (8,7) and parallel to x = -2 x = 8 All vertical lines are parallel Part VI #6: (2,2) and perpendicular to y = 3 x = 2 A vertical line is perpendicular to a horizontal line

  20. Chapter 13 WS • How can you get 100% on your final? Congrats two are locale speling be champien! http://abclocal.go.com/kgo/story?section=education&id=5360989

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