1.3 Segments and Their Measures

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Presentation transcript:

1.3 Segments and Their Measures Unit 1 Day 2

Do Now Evaluate each expression. |4 – 7| |4.3 – 1.2| √(4 + 32) √((-2)2 + 32) 3 3.1 5 √13

Using Segment Postulates In geometry, rules that are accepted without proof are called ___________, or axioms. Rules that are proved are called ___________.

Ex. 1: Finding the Distance Between Two Points Measure the length of the segment to the nearest millimeter. Use a metric ruler. Align one mark of the ruler with A. Then estimate the coordinate of B. For example if you align A with 3, B appears to align with 5.5. AB = |5.5 – 3| = |2.5| = 2.5 The distance between A and B is about 2.5 cm.

Postulate 1: Ruler Postulate A line segment can lined up with coordinates on a number line (like a ruler). The distance between points A and B is the absolute value of the difference between the coordinates of A and B. AB is also called the length of segment AB.

Ex. 2: Finding Distances on a Map Suppose the cities of Athens, Macon, and Albany, GA, lie in a line. If it is 80 miles from Athens to Macon and 90 miles from Macon to Albany, what is the distance from Athens to Albany? 80 + 90 = 170 mi.

Postulate 2: Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. (Recall: When we say a point is between two other points, it implies that the three points are ______________.) Also works for more than two segments, as long as all points collinear

Ex. 2A: Segment Addition Postulate Suppose PQ = 4.2 in., QR = 7.5 in., and PR = 11.7 in. Is Q between P and R? How do you know? Yes, since PQ + QR = PR, P must be between P and R, by the segment addition postulate

Distance Formula The distance formula is used for computing the distance between two points in a coordinate plane. If A is (x1, y1) and B is (x2, y2), then the distance between A and B is AB = _________________. √[(x2 – x1)2 + (y2 – y1)2] Relates to Pythagorean Theorem

Ex. 3 Using the Distance Formula Find the length of the segments. AB AC AD AB = √13 AC = √17 AD = √13

Congruence Segments that have the same length are call _____________ segments. Lengths are equal: AB ___ AD Segments are congruent: ĀB __ ĀD Congruent = ≅

Ex. 4: Finding Distance on a City Map On the map, the city blocks are each 340 feet apart east-west and 480 feet apart north-south. Find the walking (taxicab) distance between A and B. What would the (Euclidean) distance be if a diagonal street existed between A and B? a. To walk from A to B, you would have to walk five blocks east and three blocks north. 5 * 340 = 1700 ft. 3 * 480 = 1440 ft. Total: 1700 + 1440 = 3140 ft. b. Using the distance formula, AB = √4,963,600 ≈ 2228 ft. (912 ft. less than walking distance)

Closure In the diagram below, could you use the segment addition postulate to find the distance from D to F? Explain why or why not. D 3 cm. E You cannot use the segment addition postulate because D, E, and F are not collinear. Therefore DF is not 5 cm. long. F 2 cm.