Geometry Concepts Chapter 2 Segment Measurement and Coordinate Graphing 2.2 Find length segments (1.3) 2.3 Compare lengths of segments (1.3) 2.3 Find midpoints of segments (1.7) 2.5 Calculate coordinates of midpoint (1.7)
Section 2.2 Segments and Properties of Real Numbers Remember that there is a one-to-one correspondence between the points on the number line and the real numbers. During trout season, you must release any trout that is smaller than seven inches. Can you keep the fish below? Questions to think about: How do you find the length of a segment shown on a number line? POSTULATE: Ruler Postulate Points on a line are paired with the real numbers, and the measure of the distance between two points is the positive difference of the corresponding numbers. absolute value of the difference of their coordinates a b or b a * Examples Use the number line to find each measure. (1.) BE (2.) CF (3.) AD (4.) BG Page 2 of 10
Definition of Betweenness (Segment Addition Postulate) Point R is between points P and Q if and only if R, P, and Q are collinear and PR + RQ = PQ. PR + RQ = PQ Examples (5.) Points A, B, and C are collinear. If AB =12, BC = 48 and AC = 36 determine which point is between the other two. (6.) Points R, S, and T are collinear. If RS = 42, ST = 17 and RT = 25, determine which point is between the other two. (7.) Points K, L, and J are collinear. If KL = 31, JL = 16, and JK = 47 determine which point is between the other two. (8.) Points X, Y, and Z are collinear. If XY = 32, XZ = 49, and YZ = 81, determine which point is between the other two. (9.) If EG = 59, what are EF and FG? (10.) If JL = 120. What are JK and KL? E 8x - 14 F 4x + 1 G J 4x + 6 K 7x + 15 L Page 3 of 10
Reflexive Property Symmetric Property Transitive Property Properties of Equality for Real Numbers a = a 5 = 5 ½ = ½ 0.3 = 0.3 a = b then b = a a = b and b = c, then a = c sandwich Addition and Subtraction Properties Multiplication and Division Properties Substitution Property a + c = b + c and a c = b c a c = b c and a c = b c If a = b then a may be replaced by b in any equation. Examples (11.) If QS = 29 and QT = 52, find ST (12.) If PR = 27 and PT = 73, find RT (13.) If FG = 12 and FJ = 47, find GJ. (14.) If AC = 49 and AB = 14, find BC. NOTE: Measurements are made up of two parts the measure and the unit of measure (UOM). For example 9 cm, 9 is the measure and cm is the unit of measure. Page 4 of 10
Section 2.3 Congruent Segments Keep in mind when numerical expressions have the same value, you say that they are equal. When you have two segments with the same length, then the segments are segments segements. Questions to think about: How do you determine if two segments are congruent? Why is a statement such as AB DE incorrect? Describe the difference between saying that two segments are congruent and saying that two segments have equal length? When would you use each phrase? Definition of Congruent Segments Two segments are congruent if and only if they have the same length. Symbol Examples Use the number line to determine whether the statement is true or false. Explain your reasoning. (15.) DF is congruent toeg? (16.) EG is congruent tofh? (17.) DE is congruent toef? (18.) GH is congruent toef Page 5 of 10
We will be adding theorems to our definitions and postulates. Theorems are statements that can be justified by using logical reasoning. THEOREMS 2.1 Congruence of segments is reflexive. AB AB 2.2 Congruence of segments is symmetric. If AB CD, then CD AB 2.3 Congruence of segments is transitive. If AB CD and CD EF then AB EF Examples Determine whether the statement is true or false. Explain your reasoning. (19.) JK is congruent to KJ (20.) If AB CD and DC EF then AB EF. Examples Use the figure below to determine whether the statement is true or false. Explain your reasoning. (21.) DE GH (22.) EF FG Definition of Midpoint Midpoint is a unique point on every segment. The midpoint of a segment separates the segment into two segments of equal length. Therefore the two segments are congruent. A point M is the midpoint of a segment ST if and only if M is between S and T and SM = MT. Page 6 of 10
Examples (23.) In the figure, C is the midpoint ofab. Find the value of x. (24.) In the figure, W is the midpoint ofxy. Find the value of a. (25.) Q is the midpoint of PR. What are PR, QR, and PR? (26.) U is the midpoint of TV. What are TU, UV, and TV? P 6x - 7 Q 5x + 1 R T 8x + 11 U 12x - 1 V (27.) In the figure, K is the midpoint ofjl. Find the value of d. To bisect something means to split into two congruent parts. The midpoint of a segment bisects the segment because it creates two congruent segments. A point, line, ray, segment or plane can also bisect a segment as well. Page 7 of 10
Section 2.5 Midpoints A midpoint is a point that breaks the segment into two equal points. Questions to think about: How do you know whether a problem involves the Midpoint Formula on a number line or on a coordinate plane? How could you check your midpoint results? Midpoint Formula for a Number Line 2.5 On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is a+b 2 Midpoint Formula for a Coordinate Plane 2.6 On a coordinate plane, the coordinates of the midpoint of a segment whose x endpoints have coordinates ( x 1,y 1 ) and ( x 2,y 2 ) are 1+ x2 y +, 1 y2 2 2 Examples.Use the graph to answer the questions below. (28.) Find the coordinate of the midpoint RS (29.) Find the coordinate of the midpoint RT Page 8 of 10
Examples. (30.) Find the coordinates of M, the midpoint of JK, given endpoints J(2, -9) and K(8, 3). (31.) Find the coordinates of M, the midpoint of VW, given endpoints v(-4, -3) and W(6, 11) (32.) Find the coordinates of M, the midpoint of PR, given endpoints p(-5, 1) and r(2, -8). (33.) Since G(8, -9) is the midpoint of FE and the coordinate of E are (18, -21). Find the coordinates of F. (34.) Since K(-10, 17) is the midpoint of IJ and the coordinate of J are (4, 12). Find the coordinates of I. (35.) Since S(3, - ¾) is the midpoint of RT and the coordinate of T are (-2, 6). Find the coordinates of R. Page 9 of 10
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