1.3

1. 1.3 Segments and Their Measures 1 2 GOAL GOAL Use Segment Postulates Use the Distance Formula to measure distances To solve real-life problems, such as finding distances along a diagonal city street. Whatyou should learn Why you should learn it

2. 1.3 Segments and Their Measures USING SEGMENT POSTULATES 1 GOAL In geometry, rules that are accepted without proof are called or. postulates axioms Rules that are proved are called . theorems

3. The between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the of AB. names of points A B AB = x2 – x1 x1 coordinates of points x2 EXAMPLE 1 POSTULATES YOU NEED TO KNOW RULER POSTULATE The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the of the point. coordinate distance length

4. Extra Example 1 Measure the green bar on page 17 (it has the word postulate in it) to the nearest millimeter. Then measure it again, this time placing your ruler with the 2 at one end of the bar. If you understand the Ruler Postulate, you’ll get the same measurement as before. Your answer should be about 133 mm.

5. AB BC A B C AC EXAMPLE 2 SEGMENT ADDITION POSTULATE If B is between A and C, then AB + BC = AC. Also, if AB + BC = AC, then B is between A and C. (Remember: “between” implies the points are collinear. Do you see that the length of the blue and red segments added together is equal to the length of the purple segment?

6. Extra Example 2 Two friends leave their homes and walk in a straight line toward the other’s home. When they meet one has walked 425 meters and the other has walked 267 meters. How far apart are their homes? Click for a hint. 425 m 267 m Answer: The solution is 425 m + 267 m = 692 m.

7. Measure the length of BC at the top of page 18 to the nearest millimeter. • A car with a trailer has a total length of 27 feet. If the trailer has a total length of 13 feet, how long is the car? Checkpoint 1. about 25 mm 2. 14 ft

8. 1.3 Segments and Their Measures 2 GOAL USING THE DISTANCE FORMULA To find the distance between two points in a coordinate plane, we use the . Distance Formula

9. y B(x2, y2) A(x1, y1) x EXAMPLE 3 THE DISTANCE FORMULA If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is Important!!! Pay close attention to where the coordinates fit in the formula!

10. E(-3, 3) F(1, 2) 1 x G(-3, 0) 1 H(0, -1) Extra Example 3 Find the lengths of the segments. Tell whether any of the segments have the same length. Click for each answer. y None of the segments have the same length.

11. y K(-2, 2) 1 x 1 M(0, -1) L(-2, -3) Checkpoint Find the distance between each pair of points. Click for each answer.

12. M P Q N Incorrect: Of course, some segments have equal lengths. These are called _________________. congruent segments Important:Segments are NOT equal; they are congruent. Congruent segments have equal lengths. Correct: Be sure you understand this concept!

13. c b y a A(x2, y2) C B(x1, y1) x From the Ruler Postulate, we also know that BC = x2 – x1and AC = y2 –y1 . Then by substitution we know that c2 = a2 + b2. This is known as the Pythagorean Theorem. More in Chapter 9! Now let’s look again at the Distance Formula. Click to form a right triangle. (x2, y1) What are the coordinates of C? Square both sides of the distance formula: Let’s say AB= c, BC = a, and AC = b.

14. y C(0, 740) 370 x - 410 D(2050, -370) EXAMPLE 4 Extra Example 4 Study Example 4 before going on! On the map, the city blocks are 410 feet apart east-west and 370 feet apart north-south. • Find the walking distance between C and D. • Solution: • What would the distance be if a diagonal street existed between the two points? • Solution:

15. Find the diagonal distance between points E and F on the map. y 370 E(820, 0) x - 410 F(-410, -1110) Answer: Checkpoint about 1657 ft

16. QUESTIONS?