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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?

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

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The Distance Formula

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Find the distance between the two points. (-2,5) and (3,-1) (-2,5) and (3,-1) Let (x 1,y 1 ) = (-2,5) and (x 2,y 2 ) = (3,-1) Let (x 1,y 1 ) = (-2,5) and (x 2,y 2 ) = (3,-1)

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

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

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The Midpoint Formula The midpoint between the two points (x 1,y 1 ) and (x 2,y 2 ) is:The midpoint between the two points (x 1,y 1 ) and (x 2,y 2 ) is:

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

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

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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). = ( ) – 3 + 5 4 + 6 2 2, ( ) x 1 + x 2 y 1 + y 2 2 2, = (1, 5) = (1, 5) STEP 2 m = y 2 – y 1 x 2 – x 1 = 6 – 4 5 – (– 3) = 28282828 = 14141414 STEP 3 Find the slope of the perpendicular bisector: – 1m1m1m1m – = 1 1/4 1 1/4 = – 4 Calculate the slope of AB

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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

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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.First, find the midpoint of CD. (-1/2, 5/2) Now, find the slope of CD.Now, find the slope of CD. m=1 m=1 * Since the line we want is perpendicular to the given segment, we will use the opposite reciprocal slope for our equation.

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(y-y 1 )=m(x-x 1 ) or y=mx+b Use (x 1,y 1 )=(-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 y=-x+2

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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.) See page 492

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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 + 110 Perpendicular bisector of OB 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 + 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 is C (– 19, 53).

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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 = 3170 56.3 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) = 112.6 miles.

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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). midpoint is (–1, –1) STEP 2 Calculate the slope m = y 2 – y 1 x 2 – x 1 = –10 – 8 –5 – 3 –5 – 3 =–18 – 8 =94 STEP 3 Find the slope of the perpendicular bisector: – 1m1m1m1m = 4 9 – 1 = 9 4 STEP 4 Use point-slope form: y 1 = (x 1), 49 y = x or 49 13 9

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What is the distance formula equation?What is the distance formula equation? How do you use the distance formula to classify a triangle?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?What is the midpoint formula? How do you write the equation for a perpendicular bisector given two points?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

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8.1 Assignment p.493 3-36 every 3 rd problem

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