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– 5 + ( – 1 ) 1 + 6

Find the midpoint of the line segment joining (–5, 1) and (–1, 6). ( ). x 1 + x 2 y 1 + y 2. ,. 2. 2. 72. = ( – 3, ). ,. ( ). – 5 + ( – 1 ) 1 + 6. =. 2. 2. EXAMPLE 3. Find the midpoint of a line segment. SOLUTION.

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– 5 + ( – 1 ) 1 + 6

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  1. Find the midpoint of the line segment joining (–5, 1) and (–1, 6). ( ) x1 + x2y1 + y2 , 2 2 72 = (– 3, ) , ( ) – 5 + (–1) 1 + 6 = 2 2 EXAMPLE 3 Find the midpoint of a line segment SOLUTION Let( x1, y1 ) = (–5, 1)and( x2, y2 ) = (–1, 6 ).

  2. Write an equation for the perpendicular bisector of the line segment joining A(– 3, 4) and B(5, 6). STEP1 ( ) x1 + x2y1 + y2 , Find the midpoint of the line segment. 2 2 , ( ) – 3 + 5 4 + 6 = 2 2 EXAMPLE 4 Find a perpendicular bisector SOLUTION = (1, 5)

  3. Calculate the slope ofAB 6 – 4 = m = y2 – y1 = 1m 14 28 = 5 – (– 3) x2 – x1 STEP3 Find the slope of the perpendicular bisector: 1 1/4 – – = EXAMPLE 4 Find a perpendicular bisector STEP2 = – 4

  4. STEP4 Use point-slope form: An equation for the perpendicular bisector ofABisy = – 4x + 9. y = – 4x + 9. or ANSWER EXAMPLE 4 Find a perpendicular bisector y –5= – 4(x –1),

  5. Many scientists believe that an asteroid slammed into Earth about 65 million years ago on what is now Mexico’s Yucatan peninsula, creating an enormous crater that is now deeply buried by sediment. Use the labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate plane represents 1 mile.) EXAMPLE 5 Solve a multi-step problem Asteroid Crater

  6. STEP 1 Write equations for the perpendicular bisectors of AOand OBusing the method of Example 4. Perpendicular bisector of AO Perpendicular bisector of OB EXAMPLE 5 Solve a multi-step problem SOLUTION y = – x + 34 y = 3x + 110

  7. STEP 2 Find the coordinates of the center of the circle, where AOand OBintersect, by solving the system formed by the two equations in Step 1. EXAMPLE 5 Solve a multi-step problem y= – x + 34 Write first equation. 3x + 110 = – x + 34 Substitute for y. 4x = – 76 Simplify. x = – 19 Solve for x. y = – (– 19) + 34 Substitute the x-value into the first equation. y = 53 Solve for y. The center of the circle isC (– 19, 53).

  8. STEP 3 Calculate the radius of the circle using the distance formula. The radius is the distance between Cand any of the three given points. OC = (–19 – 0)2 + (53 – 0)2 = 3170 56.3 ANSWER The crater has a diameter of about2(56.3) = 112.6miles. EXAMPLE 5 Solve a multi-step problem Use(x1, y1) = (0, 0)and(x2, y2) = (–19, 53).

  9. 12 2 ( ) x1 + x2y1 + y2 , 2 2 42 ( ) = (– , ) 0 + (– 4) 0 + 12 , = 2 2 for Examples 3, 4 and 5 GUIDED PRACTICE For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector. 3. (0, 0), (24, 12) SOLUTION Let(x1, y1 ) = (0, 0)and( x2, y2 ) = (– 4, 12). = (–2, 6)

  10. STEP1 12 – 0 Find the midpoint of the line segment. m = y2 – y1 = –4 – 0 x2 – x1 ( ) x1 + x2y1 + y2 , , 0 + (–4) 0 + 2 ( ) = 2 2 2 2 12 STEP2 –4 Calculate the slope = EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE SOLUTION = (–2, 6) = –3

  11. STEP4 Use point-slope form: An equation for the perpendicular bisector ofABis y = x + . or 1m = = 1 1 20 1 1 20 STEP3 3 3 3 3 3 3 Find the slope of the perpendicular bisector: – 1 y 6= (x +2), – 3 – ANSWER y = x + . EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE

  12. ( ) x1 + x2y1 + y2 , 2 2 ( ) – 2 + 4 1 + (–7) , = 2 2 for Examples 3, 4 and 5 GUIDED PRACTICE For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector. 4. (–2, 1), (4, –7) SOLUTION Let(x1, y1 ) = (–2, 1)and( x2, y2 ) = (4, – 7). = (1 , –7) midpoint is ( 1, –3)

  13. –8 6 STEP1 –7 – 1 Find the midpoint of the line segment. m = y2 – y1 = 4 – (–2) x2 – x1 ( ) x1 + x2y1 + y2 , ( ) – 2 + 4 1 + (–7) , = 2 2 4 2 2 3 STEP2 Calculate the slope = = EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE SOLUTION = (1 , –7)

  14. STEP4 Use point-slope form: An equation for the perpendicular bisector ofABis y = x + . or – 1 1m = = 3 3 3 15 3 15 STEP3 4 4 4 4 4 4 Find the slope of the perpendicular bisector: y +7= (x 1), – ANSWER 4 3 y = x + . EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE

  15. ( ) x1 + x2y1 + y2 , 2 2 ( ) 3 + (– 5) 8 + (–10) , = 2 2 for Examples 3, 4 and 5 GUIDED PRACTICE For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector. 5. (3, 8), (–5, –10) SOLUTION Let(x1, y1 ) = (3, 8)and( x2, y2 ) = (– 5, –10). = (–1 , –1) midpoint is (–1 , –1)

  16. STEP1 –10 – 8 Find the midpoint of the line segment. m = y2 – y1 = –5 – 3 x2 – x1 ( ) x1 + x2y1 + y2 , ( ) 3 + (– 5) 8 + (–10) , = 2 2 9 2 2 4 STEP2 Calculate the slope –18 = = – 8 EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE SOLUTION = (–1 , –1)

  17. STEP4 Use point-slope form: An equation for the perpendicular bisector ofABis y = x  . y = x  . or – 1 1m = = 4 13 4 4 4 13 STEP3 9 9 9 9 9 9 Find the slope of the perpendicular bisector: y 1= (x 1), – 9 4 EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE ANSWER

  18. C A(16, 8) B(0, 0) x Q(6, –2) EXAMPLE 4 for Examples 3, 4 and 5 GUIDED PRACTICE The points (0, 0),(6, 22), and (16, 8) lie on a circle. Use the method given in Example 5 to find the diameter of the circle. 6. SOLUTION 20

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