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8.1 The Distance and Midpoint Formulas

8.1 The Distance and Midpoint Formulas. p. 490 What is the distance formula? How do you use the distance formula to classify a triangle? What is the midpoint formula? How do you write the equation for a perpendicular bisector given two points?. Geometry Review!.

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8.1 The Distance and Midpoint Formulas

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  1. 8.1 The Distance and Midpoint Formulas p. 490 What is the distance formula? How do you use the distance formula to classify a triangle? What is the midpoint formula? How do you write the equation for a perpendicular bisector given two points?

  2. Geometry Review! • What is the difference between the symbols AB and AB? Segment AB The length of Segment AB

  3. The Distance Formula

  4. Find the distance between the two points. • (-2,5) and (3,-1) • Let (x1,y1) = (-2,5) and (x2,y2) = (3,-1)

  5. Classify the Triangle using the distance formula (as scalene, isosceles or equilateral)

  6. RS = (5 – (–1)2 + (2 – 3)2 = 36 = 6 R –1,3 = 20 = 25 ST = (3 – 5)2 + (6 – 2)2 = 5 TR = (–1 –(–3)2 + (3 – 6)2 S 5,2 T 3,6 ANSWER Because RS ≠ ST ≠ TR, so RST is an scalene triangle. 2. The vertices of a triangle are R(– 1, 3), S(5, 2), and T(3, 6). Classify ∆RSTas scalene, isosceles, or equilateral. SOLUTION

  7. The Midpoint Formula • The midpoint between the two points (x1,y1) and (x2,y2) is:

  8. Find the midpoint of the line segment joining (–5, 1) and (–1, 6). SOLUTION Let( x1, y1 ) = (–5, 1)and( x2, y2 ) = (– 1, 6 ).

  9. Find the midpoint of the segment whose endpoints are (6,-2) & (2,-9)

  10. Write an equation for the perpendicular bisector of the line segment joining A(– 3, 4) and B(5, 6). Calculate the slope ofAB 6 – 4 = m = y2 – y1 = 14 1m 28 = 5 – (– 3) STEP1 Find the midpoint of the line segment. x2 – x1 ( ) x1 + x2y1 + y2 , 2 2 STEP3 Find the slope of the perpendicular bisector: 1 1/4 , ( ) – 3 + 5 4 + 6 = 2 2 – – = SOLUTION = (1, 5) STEP2 = – 4

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

  12. Write an equation in slope-intercept form for the perpendicular bisector of the segment whose endpoints are C(-2,1) and D(1,4). • First, find the midpoint of CD. (-1/2, 5/2) • Now, find the slope of CD. m=1 * Since the line we want is perpendicular to the given segment, we will use the opposite reciprocal slope for our equation.

  13. (y-y1)=m(x-x1) or y=mx+b Use (x1 ,y1)=(-1/2,5/2) and m=-1 (y-5/2)=-1(x+1/2) or 5/2=-1(-1/2)+b y-5/2=-x-1/2 or 5/2=1/2+b y=-x-1/2+5/2 or 5/2-1/2=b y=-x+2 or 2=b y=-x+2

  14. 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.) Asteroid Crater See page 492

  15. 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. STEP 1 Write equations for the perpendicular bisectors of AOand OBusing the method of Example 4. Perpendicular bisector of AO Perpendicular bisector of OB y = – x + 34 y = 3x + 110 Write first equation. y = – x + 34 Substitute for y. 3x + 110 = – x + 34 Simplify. 4x = – 76 Solve for x. x = – 19 Substitute the x-value into the first equation. y = – (– 19) + 34 y = 53 Solve for y. The center of the circle isC (– 19, 53).

  16. 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. Use(x1, y1) = (0, 0)and(x2, y2) = (–19, 53).

  17. Use point-slope form: STEP4 –10 – 8 y = x  or m = y2 – y1 – 1 1m = = –5 – 3 = x2 – x1 ( ) x1 + x2y1 + y2 , 13 2 2 4 9 4 4 9 9 4 9 9 STEP3 Find the slope of the perpendicular bisector: y  1 =  (x  1), STEP2 Calculate the slope – ( ) 3 + (– 5) 8 + (–10) , = 2 2 –18 9 = = 4 – 8 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). midpoint is (–1 , –1)

  18. What is the distance formula equation? • How do you use the distance formula to classify a triangle? The distance formula will tell you the length of the sides of the triangle. (2= isosceles, 3=equilateral) • What is the midpoint formula? • How do you write the equation for a perpendicular bisector given two points? Use the points to find the slope, use the negative reciprocal, use the midpoint formula to find a the middle point and use y = mx+b to write your equation

  19. 8.1 Assignment • 493 3-36 every 3rd problem

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