ࡱ> IKHq` ,zbjbjqPqP ;~::]\  2228$3(4l L24>7:x7x7x7$~htE ӂx7x7[K^Lx7 x7/|  _x74 k20LZZ$_Z _<^L   $0$   0    Geometry Lesson Notes 1.3A Date ________________ Objective: Find the distance between two points. Distance between two points: the measure of the line segment between the points. When two points are on a number line: To find AB, the measure of  EMBED Equation.DSMT4 , with coordinates a and b on the number line, take the difference of the values. d = a b or b a The measure of a segment is always positive, so take the absolute value of the difference. (Note: The absolute value of the difference is not always the same as the difference of the absolute values.) Example 1 (p 21): Find Distance on a Number Line  Find AB for different values of A and B.  Find the distance between A(6, 3) and B(6, (5). Find CD, given C((4, 7) and D(1, 7). Find the measure of  EMBED Equation.DSMT4 . (Recall the Pythagorean Theorem.) For two points on the coordinate plane: Distance Formula: The distance, d, between two points with coordinates (x1, y1) and (x2, y2) is given by the formula:  EMBED Equation.3  Example 2 (p 21): Find Distance on a Coordinate Plane Find ST, the length of the segment with endpoints S((3, 4) and T(6, 1). ( HW: A6 pp 25-27 #13-18, 19-29 odd, 30 Geometry Lesson Notes 1.3B Date ________________ Objective: Find the midpoint of a segment. Midpoint of a segment: the midpoint, M, of  EMBED Equation.DSMT4  is the point between P and Q such that PM = MQ. Note: If M is between P and Q, then P and M and Q must be collinear. On a number line, The coordinate of the midpoint of the line segment whose endpoints have the coordinates a and b is given by the formula:  EMBED Equation.3  Example 3a (p23): Find Coordinates of the Midpoint Try it on the number line below.  EMBED Equation.3  Find point b on the number line if the midpoint of the segment is 15 and the value of point a is (20. Plot the points A(4, (3) and B(8, (3). Draw  EMBED Equation.DSMT4 . What is the midpoint of the line segment? Plot the points C(0, 8) and D((7, 8). Draw EMBED Equation.DSMT4 . What is the midpoint of the line segment? Draw  EMBED Equation.DSMT4 . What is the midpoint of the line segment? On the coordinate plane, The coordinates of the midpoint, M, of a line segment whose endpoints have the coordinates (x1, y1) and (x2, y2) are given by the formula:  EMBED Equation.DSMT4  Example 3b (p23): Find Coordinates of the Midpoint: Find the midpoint of  EMBED Equation.DSMT4  for A(4, (3) and D((7, 8). Find the coordinates of the midpoint of  EMBED Equation.DSMT4  for G(8, "6) and H("14, 12). Example 4 (p 23): Find the coordinates of the endpoint! Find the coordinates of D if E("6, 4) is the midpoint of  EMBED Equation.DSMT4  and F has coordinates ("5, "3). NOTE: THIS IS AN EXAMPLE OF USING A FORMULA IN REVERSE! IMPLICATION: What does it mean to be a midpoint? What must be true? The midpoint of a segment splits the segment into two congruent segments. What is the implication if two segments are congruent? NOTE: YOU WILL USE IMPLICATIONS TO WRITE EQUATIONS TO DESCRIBE GEOMETRIC RELATIONSHIPS! Example 5 (p 23): Use Algebra to Find Measures What is the measure of  EMBED Equation.DSMT4  if Q is the midpoint of  EMBED Equation.DSMT4 ? Use the implication of what it means to be a midpoint to solve the problem!  A B 4 C 4 D 9 Practice: Given S((3, 4) and T(6, 1). Find the midpoint of  EMBED Equation.DSMT4 . * Given M((30, 2) is the midpoint of  EMBED Equation.DSMT4  and K(12, 11), find J. * Given S is the midpoint of  EMBED Equation.DSMT4 ,  EMBED Equation.3 , and  EMBED Equation.3 . What does it mean to say that S is the midpoint of  EMBED Equation.DSMT4 ? What must be true? Draw a sketch. Find x and the measure of  EMBED Equation.DSMT4 . * Given U is between T and V,  EMBED Equation.3 , TU = 9, and  EMBED Equation.3 . Find x and determine if U is the midpoint of  EMBED Equation.DSMT4 . (Important skill!) Segment Bisectors  Plot two points Q and R on the number line and then draw  EMBED Equation.DSMT4  through M, the midpoint of  EMBED Equation.DSMT4 . Segment bisector: any segment, line, or plane that intersects a segment at its midpoint. IMPLICATION: A segment bisector creates two congruent segments. REMEMBER: Objects and figures are congruent. e.g.  EMBED Equation.DSMT4  Numbers and measures are equal. e.g. AB = BC Practice: In the figure,  EMBED Equation.DSMT4  bisects  EMBED Equation.DSMT4  at X and  EMBED Equation.DSMT4  bisects  EMBED Equation.DSMT4  at Y. Given the following conditions, find the value of x and the measure of the indicated segment. a. AX = 2x + 11, XB = 4x 5;  EMBED Equation.DSMT4  b. AB = x + 3, AX = 3x 1;  EMBED Equation.DSMT4  c. 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