– 5 + (– 1) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, Find the midpoint of a line segment EXAMPLE 3 Let ( x 1, y 1 ) = (–5, 1) and ( x 2, y 2 ) = (– 1, 6 ). = ( – 3, ) 7272 Find the midpoint of the line segment joining (–5, 1) and (–1, 6). SOLUTION

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

Find a perpendicular bisector EXAMPLE 4 STEP 2 m = y 2 – y 1 x 2 – x 1 = 6 – 4 5 – (– 3) = 2828 = 1414 STEP 3 Find the slope of the perpendicular bisector: – 1m1m – = 1 1/4 = – 4 Calculate the slope of AB

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

Solve a multi-step problem EXAMPLE 5 Asteroid Crater 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.)

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

Solve a multi-step problem EXAMPLE 5 STEP 2 Find the coordinates of the center of the circle, where AO and OB intersect, by solving the system formed by the two equations in Step 1. y = – x + 34 Write first equation. 3x = – 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 is C (– 19, 53).

Solve a multi-step problem EXAMPLE 5 STEP 3 Calculate the radius of the circle using the distance formula. The radius is the distance between C and any of the three given points. OC = (–19 – 0) 2 + (53 – 0) 2 = Use (x 1, y 1 ) = (0, 0) and (x 2, y 2 ) = (–19, 53). ANSWER The crater has a diameter of about 2(56.3) = miles.

GUIDED PRACTICE for Examples 3, 4 and 5 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 0 + (– 4) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, Let (x 1, y 1 ) = (0, 0) and ( x 2, y 2 ) = (– 4, 12). = ( –, ) = (–2, 6)

EXAMPLE 4 SOLUTION STEP 1 Find the midpoint of the line segment. 0 + (–4) = ( ) 2 2, x 1 + x 2 y 1 + y 2 2 2, = (–2, 6) STEP 2 Calculate the slope m = y 2 – y 1 x 2 – x 1 = 12 – 0 –4 – 0 = 12 –4 = –3 GUIDED PRACTICE for Examples 3, 4 and 5

EXAMPLE 4 STEP 3 Find the slope of the perpendicular bisector: – 1m1m – 1 – 3 = = 1 3 GUIDED PRACTICE for Examples 3, 4 and 5 STEP 4 Use point-slope form: y  6 = (x + 2), 1 3 y = x +. or ANSWER An equation for the perpendicular bisector of AB is y = x

GUIDED PRACTICE for Examples 3, 4 and 5 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 – (–7) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, Let (x 1, y 1 ) = (–2, 1) and ( x 2, y 2 ) = (4, – 7). = (1, –7) midpoint is ( 1, –3)

EXAMPLE 4 SOLUTION STEP 1 Find the midpoint of the line segment. ( ) x 1 + x 2 y 1 + y 2 2 2, STEP 2 Calculate the slope m = y 2 – y 1 x 2 – x 1 = –7 – 1 4 – (–2) GUIDED PRACTICE for Examples 3, 4 and 5 – (–7) = ( ) 2 2, = (1, –7) = 6 –8 = 4 3

EXAMPLE 4 STEP 3 Find the slope of the perpendicular bisector: – 1m1m GUIDED PRACTICE for Examples 3, 4 and 5 = 3 4 – 1 = 4 3 STEP 4 Use point-slope form: y + 7 = (x  1), 3 4 y = x +. or ANSWER An equation for the perpendicular bisector of AB is y = x

GUIDED PRACTICE for Examples 3, 4 and 5 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 3 + (– 5) 8 + (–10) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, Let (x 1, y 1 ) = (3, 8) and ( x 2, y 2 ) = (– 5, –10). = (–1, –1) midpoint is (–1, –1)

EXAMPLE 4 SOLUTION STEP 1 Find the midpoint of the line segment. STEP 2 Calculate the slope m = y 2 – y 1 x 2 – x 1 = –10 – 8 –5 – 3 = –18 – 8 GUIDED PRACTICE for Examples 3, 4 and (– 5) 8 + (–10) ( ) = 2 2, x 1 + x 2 y 1 + y 2 2 2, = (–1, –1) = 9 4

EXAMPLE 4 STEP 3 Find the slope of the perpendicular bisector: – 1m1m GUIDED PRACTICE for Examples 3, 4 and 5 = 4 9 – 1 = 9 4 STEP 4 Use point-slope form: y  1 =  (x  1), 4 9 y =  x . or ANSWER An equation for the perpendicular bisector of AB is y =  x 

EXAMPLE 4 GUIDED PRACTICE for Examples 3, 4 and 5 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. Q(6, –2) x A(16, 8) C B(0, 0) SOLUTION 20