1.2 Use Segments and Congruence

The Ruler Postulate The distance between two points on a line is the absolute value of the difference of the coordinates of the two points. AB = |x2 – x1| A B x1 x2

Segment Addition Postulate If B is between A and C then AB + BC = AC AC AB BC A B C *Betweenness: by definition, a point B is between two other points A and C if all three points are collinear and AB + BC = AC.

Congruent Segments Two segments are congruent if they have the same length. A 12 B C D 12 AB is congruent to CD AB = CD ~

Explain what MN means and what MN means. If PQ = 4 and QN = 5, what is PN? If MP = 3 and MN = 14, what is PN? M P Q N MN is the line segment itself MN means the length of the line segment PN = PQ + QN = 4 + 5 = 9 PN = MN – MP = 14 – 3 = 11

Plot the points then tell whether the line segments are congruent. 1. A(0, 1), B(4,1), C(1,2), D(1,6) AB & CD? 2. J(-6,-8), K(-6,-2), L(-2,-4), M(-6,-4) JK & LM? AB = 4 – 0 = 4 CD = 6 - 2 = 4 AB = CD ~ JK = -2 – (-8) = -2 + 8 = 6 LM = -6 – (-2)= -6 + 2 = |-4| = 4 ~ JK ≠ LM

1.3: The Distance and Midpoint Formulas

Segment Terminology Midpoint – the point that divides a segment into two congruent segments. Bisect – “to cut in half” Segment Bisector – a segment, ray, line, or plane that intersects a segment at its midpoint.

The Distance Formula Used to find the distance between two points

Example Find the distance between A(4,8) and B(1,12) A (4, 8) B (1, 12)

Apply The Distance Formula A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a goal line. The ball lands 45 yards from the same goal line and 40 yards from the same sideline. How far was the ball kicked?

Solution The ball is kicked from the point (10, 5) and lands at the point (40, 45). d = (40 – 10) 2 + (45 – 5) 2 = 900 + 1600 = 2500 = 50

The Midpoint Formula Used to find the center point (bisector) of a line segment.

Example Find the midpoint between A(4,8) and B(1,12) A (4, 8) B (1, 12)

Apply the Midpoint Formula You are using computer software to design a video game. You want to place a buried treasure chest halfway between the center of the base of a palm tree and the corner of a large boulder. At what coordinate should you place the treasure chest? (25, 175) (112.5, 125) (200, 75)

Solution ( ) ( ) There are two coordinates on this map: (25, 175) There are two coordinates on this map: Palm tree (200, 75) boulder (25, 175) Use the midpoint formula to solve for the point halfway between the two landmarks. (112.5, 125) (200, 75) 25 + 200 2 ( ) 175 + 75 , 225 2 ( ) 250 , = = (112.5, 125)

Your Turn! Solve for the distance between the following points: (2, 7) and (11, 9) (-5, 8) and (2, -4)

Your Turn! Solve for the midpoint between the following points: (2, 7) and (14, 9) (-5, 8) and (2, -4)