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Geometry

Geometry. Measuring Segments Section 1-4. This section shows us how to measure line segments . We will establish a lot of basic rules associated with line segments and armed with that knowledge, we will expand those rules into the plane and eventually into space.

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Geometry

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  1. Geometry Measuring Segments Section 1-4

  2. This section shows us how to measure line segments. We will establish a lot of basic rules associated with line segments and armed with that knowledge, we will expand those rules into the plane and eventually into space. Everything that is true about one dimensional figures (i.e., the line) will be applicable to two dimensional figures on a plane. The measurement of line segments ties geometry to algebra. We will learn that the measurement of line segments is directly related to the congruence of geometric figures. …\GeoSec01_04.ptt

  3. NOTE that segment A to C is written AC. The concept of between. Point D is NOTcollinear with points A and C and is NOTbetween points A and C. Given a line and two points on that line. D • • C • B A • If point B is collinear with points A and C, then point B is between point A and point C. …\GeoSec01_04.ptt

  4. The measure of segment AC is defined as the distance between points A and C. Thus, the measure of a segment is the distance between its endpoints. • C A • There are different measures that we can use. Yards, feet, meters, centimeters, kilometers are a few examples. These different measures are called units of measure. …\GeoSec01_04.ptt

  5. xy •• -3 -2 -1 0 1 2 3 4 5 6 Postulate 1-1, The Ruler Postulate; The points on any line can be paired with the real numbers so that, given any two points Pand Q on the line, Pcorresponds to zero, and Q corresponds to a positive number. This postulate states that the measure of a segment is always positive and that any segment can be aligned with a number line. | -2 - 5 | = distance between point x and point y. …\GeoSec01_04.ptt

  6. • R Q • P • Postulate 1-2, Segment Addition Postulate; - IfQ is betweenPand R, thenPQ+ QR = PR. IfPQ+ QR = PR, thenQ is betweenPand R. If we have three collinear points, with one point between two other points, then the segment can be separated into two parts that equal the length of the whole segment. What else does it state? …\GeoSec01_04.ptt

  7. • R Q • P • Postulate 1-2, Segment Addition Postulate; - IfQ is betweenPand R, thenPQ+ QR = PR. IfPQ+ QR = PR, thenQ is betweenPand R. If two small segments are equal to a larger segment and each segment has a point in common, then the common point is between the other two points. Example: find LM if L is betweenN and M, NL = 6x-5, LM = 2x+3, and NM = 30. …\GeoSec01_04.ptt

  8. 30 6x - 5 2x + 3 • N • L • M First, DRAW A PICTURE. Using Algebra we can now solve for the unknown, x, and then determine the length of LM. (6x - 5)+(2x + 3)=30 8x - 2 =30 8x=32 x=4 …\GeoSec01_04.ptt

  9. The side opposite the right angle, in this case sidec, is called the hypotenuse. c b a Pythagorean Theorem- In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse ( a2 + b2 = c2 ). Sides a and b are the sides that forms the right angle,which is defined as a 90o angle. We accept this without proof at this time. You have learned and used this theorem in Algebra I and it should be no surprise how it operates. …\GeoSec01_04.ptt

  10. Distance Formula - The distance, d, between any two points (x1, y1) and (x2, y2) is given by the formula y d=  (x2 -x1 )2 + (y2 -y1 )2 (4,4) 5 4 3 2 1 0 -1 -2 -3 -4 -5 x (-3- 4) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 (-5,-3) (-5 - 4) A right triangle The distance between two points on a coordinate plane represents a right triangle and the distance between those points is the hypotenuse of that right triangle. …\GeoSec01_04.ptt

  11. y d=  (x2 -x1 )2 + (y2 -y1 )2 (4,4) 5 4 3 2 1 0 -1 -2 -3 -4 -5 x (-3- 4) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 (-5,-3) (-5 - 4) It is the Pythagorean Theorem applied to the coordinate plane! …\GeoSec01_04.ptt

  12. Distance Formula - The distance, d, between any two points (x1, y1) and (x2, y2) is given by the formula d=  (x2 -x1 )2 + (y2 -y1 )2 d=  (5 -(-2) )2 + (8 - 3 )2 d=  (-2 -5 )2 + (3 - 8 )2 d=  (7 )2 + (5 )2 d=  (-7 )2 + (-5 )2 d=  49 + 25 d=  49 + 25 d=  74 d=  74 Note that P1 = (x1,y1) and P2 = (x2,y2) are two different points on the x-y coordinate plane. If you are given two points, say (-2, 3) and (5, 8), it does not matter which one of these points is assigned to P1 and which is assigned to P2. The same …\GeoSec01_04.ptt

  13. Summary: In this section we have learned, through The Ruler Postulate, how to algebraically measure the distancebetween two points on a number line and on the x-y coordinate plane using several formulas. These formulas are the absolute value of the difference of two points on a number line and the distanceformula for the coordinate plane. We have revisited the Algebra I Pythagorean Theorem, which serves as the basis for the distance formula. …\GeoSec01_04.ptt

  14. END OF LINE …\GeoSec01_04.ptt

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