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1-2 Measuring Segments

This guide explores the concepts of measuring segments, finding midpoints, and calculating distances in the coordinate plane. It covers essential vocabulary such as segments, midpoints, bisectors, and congruence. Step-by-step examples illustrate how to apply the distance and midpoint formulas, providing clear instructions for solving problems involving segments. By mastering these concepts, students will enhance their ability to navigate geometric relationships and calculations in two-dimensional space.

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1-2 Measuring Segments

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  1. 1-2 Measuring Segments Objectives Use length and midpoint of a segment. Apply distance and midpoint formula.

  2. Vocabulary coordinatemidpoint distancebisect lengthsegmentbisectorcongruent segments

  3. A point corresponds to one and only one number (or coordinate) on a number line.

  4. A B AB = |a – b| or |b - a| a b Distance (length): the absolute value of the difference of the coordinates.

  5. Example 1: Finding the Length of a Segment Find each length. A. BC B. AC BC = |1 – 3| AC = |–2 – 3| = |– 2| = |– 5| = 2 = 5

  6. Point B is betweentwo points A and Cif and only if all three points are collinear and AB + BC = AC. A

  7. bisect: cut in half; divide into 2 congruent parts. midpoint: the point that bisects, or divides, the segment into two congruent segments

  8. 4x + 6 = 7x - 9 +9 +9 4x + 15 = 7x -4x -4x 15 = 3x 3 5 = x

  9. It’s Mr. Jam-is-on Time!

  10. Recap! 1. M is between N and O. MO = 15, and MN = 7.6. Find NO. 2.S is the midpoint of TV, TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV. 3. LH bisects GK at M. GM =2x + 6, and GK = 24.Find x.

  11. 1. M is between N and O. MO = 15, and MN = 7.6. Find NO.

  12. 2. S is the midpoint of TV, TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV.

  13. 3.LH bisects GK at M. GM =2x + 6, and GK = 24.Find x.

  14. 1-6 Midpoint and Distance in the Coordinate Plane

  15. Vocabulary • Coordinate plane: a plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. y-axis The location, or coordinates, of a point is given by an ordered pair (x, y). II I x-axis III IV

  16. Midpoint Formula The midpoint M of a AB with endpoints A(x1, y1) and B(x2, y2) is found by

  17. Example Find the midpoint of GH with endpoints G(1, 2) and H(7, 6).

  18. Example M(3, -1) is the midpoint of CD and C has coordinates (1, 4). Find the coordinates of D.

  19. Distance Formula The distance d between points A(x1, y1) and B(x2, y2) is

  20. Example Use the Distance Formula to find the distance between A(1, 2) and B(7, 6).

  21. Pythagorean Theorem In a right triangle, a2 + b2 = c2 c is the hypotenuse (longest side, opposite the right angle) a and b are the legs (shorter sides that form the right angle)

  22. Example Use the Pythagorean Theorem to find the distance between J(2, 1) and K(7 ,7).

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