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Chapter 1 – Discovering Points, Lines, Planes, and Angles

Chapter 1 – Discovering Points, Lines, Planes, and Angles. Chapter 1.4: Using Technology – Using formulas – Measuring Segments. Objectives. To find the distance between two points on a number line and between two points in a coordinate plane

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Chapter 1 – Discovering Points, Lines, Planes, and Angles

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  1. Chapter 1 – Discovering Points, Lines, Planes, and Angles Chapter 1.4: Using Technology – Using formulas – Measuring Segments

  2. Objectives • To find the distance between two points on a number line and between two points in a coordinate plane • To use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle

  3. Vocabulary • between • measure • Postulate

  4. The points of a line can be put into one–to–one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. AB Ruler Postulate A B a b Coordinate of A Coordinate of B AB = a – b 4cm 1cm The measure of the length of the segment AB (It’s a number)

  5. Geometry vs Algebra • Segments are Congruent • Symbol [  ] • AB  CD • Lengths of segments are equal. • Symbol [ = ] • AB = CD

  6. Example 1 A C B E -8 -6 -4 -2 0 2 4 6 8 AC = = = = AB = = = CE = = =

  7. Practice XY = | | = | | = ZW = | | = | | = 4 Because XY = ZW, XYZW.

  8. Segment Addition Postulate • If 3 points A, B, & C are collinear & B is between A & C, then C B A

  9. AN = 2x – 6 = 2(__) – 6 = ____ NB = x + 7 = (__) + 7 = ____ Substitute 8 for x. Segment Addition Postulate If AB = 25, find the value of x. Then find AN and NB. Use the Segment Addition Postulate to write an equation. AN + NB = ABSegment Addition Postulate ( ) + ( ) = 25 Substitute. = 25 Simplify the left side. = Subtract 1 from each side. = Divide each side by 3. AN = ___ and NB = ___, which checks because the sum of the segment lengths equals 25. 1-4

  10. Practice Find MN if LN = 20 LM = 15 Find DS if DT = 60 DS = 2x – 8 ST = 3x – 12 L M N 15 LM + MN = LN 15 + MN = 20 MN = 20 D S T 2x - 8 3x - 12 60

  11. The Pythagorean Theorem is probably the most famous mathematical relationship. It states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a2 + b2 = c2

  12. Find the distance between A (1,5) or (x1,y1) and B (-2,1) or (x2, y2) BC = |1 – (-2)| = AC = |5 – 1| = a = b = a2 + b2 = c2 The distance (d) or c is: Distance Formula using Pythagorean Theorem A B C d = √(x2 – x1)2 + (y2 – y1)2

  13. Practice • Find PQ for P (-3, -5) and Q (4, -6)

  14. Exit Slip • What is the distance formula? • What is the Pythagorean Theorem?

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