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Happy Chinese New Year Present

Happy Chinese New Year Present. Ms Wathall Coordinate Geometry C2. Coordinate geometry. A) Midpoint of a line B) Distance between two points C) Equation of a circle after CNY. Midpoints. To find the midpoint between two points we find the average of the x’s and the average of the y’s.

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Happy Chinese New Year Present

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  1. Happy Chinese New Year Present Ms Wathall Coordinate Geometry C2

  2. Coordinate geometry A) Midpoint of a line B) Distance between two points C) Equation of a circle after CNY

  3. Midpoints • To find the midpoint between two points we find the average of the x’s and the average of the y’s. • To find the midpoint between (x1,y1) and (x2,y2) we use x = x1+ x2 y = y1 +y2 2 2

  4. Example 1 • Find the midpoint between the points (-3,4) and (-1,6) • Answer (-2,5)

  5. Centre of a circle • If we know that two points, A and B are a diameter of a circle then the centre of the circle is the midpoint of A and B! A B

  6. Example 2 • The line AB is a diameter of a circle where A and B are (-4,6) and (7,8). Find the coordinates of the centre. • Answer (3/2, 7) • Please complete ex 4A

  7. Perpendicular lines • Perpendicular lines look like this: • Question plot the points E(-1,3) G(6,1) H(-3,-5) and F (5,-6) • Join EF and GH and find their respective gradients • Multiply these two gradients together. • What do you notice?

  8. Perpendicular Lines • Two lines are perpendicular if there is a 900 angle between them. • It looks like this : what is the relationship between their slopes? • So m1 X m2 = -1 • Or if two lines are perpendicular one gradient is the negative reciprocal of the other.

  9. Example 3 • Triangle ABC has vertices A(4, 5), B(8, 13) and C(-4, 9). Use the slopes to show that triangle ABC is a right angle.

  10. A handy theorem • The perpendicular from the centre of a circle to a chord bisects the chord

  11. Example 4 • The line AB is a diameter of a circle centre C where A and B are (-2,5) and (2,9). The line l passes through C and is perpendicular to FG. Find the equation of l. • Here draw a diagram! • Follow these steps: • 1) Find the midpoint of AB • 2) Find the gradient of AB and take negative reciprocal • 3) Use y-y1 = m(x-x1) • Ex 4B l B A

  12. Distance between two points • Look at this Q(7,5) y 2 units 5 units P(2,3) x So using Pythagoras PQ2 = 52+22

  13. Distance between two points • Look at this Q(x2,y2) y (y2-y1) units P(x1,y1) (x2-x1)units x So using Pythagoras PQ2 = (x2-x1)2 + (y1-y2)2

  14. Distance Formula • The formula to find the distance between the two points (x1,y1) and (x2,y2) is: (we use Pythagoras!) • PQ2 = (x1-x2)2 + (y1-y2)2 • Taking the square root of both sides gives • PQ = (x1-x2)2 + (y1-y2)2

  15. Example 5 • Find the distance between the two points (-4,3) and (2,-1) • Answer 2√13 units

  16. Another handy theorem • The angle in a semi circle is 900 • Please complete Ex 4C

  17. Finding the centre • Two perpendicular bisectors of two chords will intersect at the centre of the circle.

  18. Equation of a circle centre (0,0) • Let us look at a circle with the centre at (0,0) and whose radius is r. • Using Pythagoras we have • X2+y2 = r2 P (x,y)

  19. Equation of a circle centre (a,b) • We still use Pythagoras to find the equation of the circle with centre (a,b) • Let us look at a numerical example first • (x-2) 2 + (y-4) 2 = r 2 (x,y) (2,4)

  20. General equation of a circle • The general equation of the circle with centre (a,b) and radius r is • (x-a) 2 + (y-b) 2 = r 2

  21. Equation of a circle : centre (a, b) radius r y (x , y) r (a , b) x Equation of circle is (x – a)2 + (y – b)2 =r2

  22. An applet • Here is an applet to show the equation of circles with different centres

  23. Example 1 • Find the equation of the circle with centre (-4, 5) and radius 6

  24. Example 2 • Show that the circle (x-2) 2+ (y-5)2 = 13 passes through the point (4,8)

  25. An extra example • Find the centre and radius of the following circle • X2 + y2 -10x -2y +17 = 0 • Here use the method of completing the square • X2 -10x +25 + y2 -2y +4 = -17+29 • Can you complete this? • Click here for lots of examples and a quick quiz

  26. Another Example Find the centre and radius of the circles: • x2 + y2 -2x +4y – 9 = 0

  27. Finding points of intersection Find where the line y = x + 5 meets the circle x2 + (y- 2)2 = 29

  28. Example Show the line y = x – 7 does not meet the circle (x + 2)2 + y2 = 33

  29. More practice with CTS

  30. Example 3 • AB is a diameter of a circle where A is ( 4,7) and B is ( -8,3) . Find the equation of the circle. • Step 1 : find the radius • Step 2: Find the centre which is the midpoint of AB • Step 3: use the general equation

  31. An interesting circle theorem The angle between a tangent and the radius is 90 0 A tangent meets the circle at one point only Ex 4D

  32. Example 4 • Show that the line y = 2x-2 meets the circle (x-32)2 + (y-2) 2 = 20 at the points A and B. Show that AB is a diameter of the circle • Step 1 solve simultaneously • Step 2 find distance AB and prove it is 4√5 (why?) • Ex 4E Mixed exercise 4F

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