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# 10.1 Distance and Midpoint Formulas

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### 10.1 Distance and Midpoint Formulas

1. 1. 10.1 The Distance and Midpoint Formulas Algebra 2 Mr. Swartz
2. 2. Objectives/Standard/Assignment Objectives: 1. Find the distance between two points in the coordinate plane, and 2. Find the midpoint of a line segment in the coordinate plane.
3. 3. Geometry Review! • What is the differenceWhat is the difference between the symbols AB andbetween the symbols AB and AB?AB? Segment ABSegment AB TheThe lengthlength ofof Segment ABSegment AB
4. 4. The Distance Formula • The Distance d between the points (x1,y1) and (x2,y2) is : 2 12 2 12 )()( yyxxd −+−=
5. 5. The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a2 + b2 = c2 . Remember!
6. 6. Find the distance between the two points. • (-2,5) and (3,-1)(-2,5) and (3,-1) • Let (xLet (x11,y,y11) = (-2,5) and (x) = (-2,5) and (x22,y,y22) = (3,-1)) = (3,-1) 22 )51())2(3( −−+−−=d 3625+=d 81.761 ≈=d
7. 7. Classify the Triangle using theClassify the Triangle using the distance formula (as scalene,distance formula (as scalene, isosceles or equilateral)isosceles or equilateral) 29)61()46( 22 =−+−=AB 29)13()61( 22 =−+−=BC 23)63()41( 22 =−+−=AC Because AB=BC the triangle isBecause AB=BC the triangle is ISOSCELESISOSCELES C: (1.00, 3.00) B: (6.00, 1.00) A: (4.00, 6.00) C B A
8. 8. The Midpoint Formula • The midpoint between the twoThe midpoint between the two points (xpoints (x11,y,y11) and (x) and (x22,y,y22) is:) is: ) 2 , 2 ( 1212 yyxx m ++ =
9. 9. MIDPOINT FORMULA
10. 10. Find the midpoint of theFind the midpoint of the segment whose endpointssegment whose endpoints are (6,-2) & (2,-9)are (6,-2) & (2,-9)       −+−+ 2 92 , 2 26       − 2 11 ,4
11. 11. Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2). Substitute. Write the formula. Simplify. Additional Example 1: Finding the Coordinates of a Midpoint G(–4, 3) H(6, -2)
12. 12. Additional Example 2: Finding the Coordinates of an Endpoint Step 1 Let the coordinates of P equal (x, y). Step 2 Use the Midpoint Formula. P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q.
13. 13. Additional Example 2 Continued Multiply both sides by 2. Isolate the variables. –2 = –5 + x +5 +5 3 = x 6 = 4 + y −4 −4 Simplify. 2 = y Set the coordinates equal. Step 3 Find the x- coordinate. Find the y-coordinate. Simplify.