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Notes on Simple Distance and Pythagorean Theorem Name_________________________ On a coordinate plane, the distance between two points that lie on the same horizontal line is the absolute value of the difference in the x-coordinates. For example, on the coordinate plane to the right, point A has an xcoordinate of -2, and point B has an x-coordinate of 3. 6 4 2 A distance from A to B = 2 3 5 5 B -5 5 -2 Of course, you are also free to just count if that is easier. C A D -4 -6 On the same coordinate plane, what is the distance from point C to point D? Suppose point E has coordinates of (1.7, 3) and point F has coordinates of (-2.1, 3). What is the distance from point E to point F? --------------------------------------------------------------------------------------------------------------------Similarly, the distance between two points that lie on the same vertical line is the absolute value of the difference in the y-coordinates. 6 Suppose the locations of the post office and the restaurant are represented as the points P and R on the coordinate plane to the left. P G 4 2 -5 5 If one unit on the coordinate plane represents one mile in real life, how many miles apart is the post office from the restaurant? R -2 C -4 Since the two points lie on the same vertical line, one needs to find the absolute value of the difference in the y-coordinates. -6 distance from P to R = 6 (1) 7 7 miles Suppose the locations of the gym and the car wash are represented by points G and C, respectively. What is the distance (in miles) from the gym to the car wash? If point K lies at (-5, -6.75), and point L lies at (-5, -0.5), what is the distance (in units) from point K to point L? --------------------------------------------------------------------------------------------------------------------If two points lie on the same horizontal line, then they have the same ____________________. If two points lie on the same vertical line, then they have the same _____________________. At this point in the lesson, it will be necessary to review the Pythagorean Theorem: leg12 leg 2 2 hypotenuse 2 Remember that the two legs are the two sides which form the right angle in the right triangle. The hypotenuse is the longest side, and it is located across from the right angle. --------------------------------------------------------------------------------------------------------------------Find the length of the side represented by a variable. Express all answers as exact decimals or as radicals in simplest form. 6 h 8 n 2 4 --------------------------------------------------------------------------------------------------------------------The Pythagorean Theorem can be used to find the distance between any two points on a coordinate plane. 6 P In order to find the distance from point P to point R, connect point P to point R with a line segment. This segment will form the hypotenuse of the right triangle. 4 2 R -5 Next, draw a segment starting at point P going downward, and start a segment starting at point R going to the left. The segments should meet and stop at (1, 1). 5 -2 -4 -6 The result should be a right triangle. Find the lengths of the two legs. Finally, using the lengths of the two legs and the Pythagorean Theorem, one can find the length of the hypotenuse (which is the distance from point P to point R). --------------------------------------------------------------------------------------------------------------------S What is the distance from point S to point T on the coordinate plane to the left? Express your answer as a radical in simplest T form. 6 4 2 -5 V 5 -2 -4 W -6 What is the distance from point V to point W? Express your answer as a radical in simplest form AND as a decimal rounded to the nearest tenth. HW on Simple Distance and Pythagorean Theorem Name_________________________ Find the distance between the requested points using the coordinate plane to the right. Express all answers as whole numbers. A 1. from point A to point B 6 4 C 2. from point C to point D D 2 F B -5 3. from point E to point F 4. from point E to point G 5 -2 -4 E G -6 --------------------------------------------------------------------------------------------------------------------Find the distance between the requested points using the coordinate plane to the left. Express all answers as whole numbers or as radicals in simplest form. H M 6 5. from point H to point L 6. from point M to point N 7. from point P to point Q 4 L 2 N P -5 5 -2 -4 Q -6 --------------------------------------------------------------------------------------------------------------------Find the distance between the requested points based on their coordinates. Express all answers as whole numbers or as radicals in simplest form. It may help to sketch a coordinate plane. 8. point R: (4, 10) point S: (21, 10) 9. point T: (2, 4) point U: (4, 9) 10. point W: (-1, 3) point X: (6, 18) 11. point Y: (-8, 0.7) 12. point A: (-2, -10) 13. point C: (2, 5) point Z: (-8, -1.2) point B: (-26, -17) point D: (-4, 3) --------------------------------------------------------------------------------------------------------------------Suppose that F represents the farm, G represents the gas station, and M represents the market on the coordinate plane to the left. Also, each unit represents one mile. 6 F 4 2 -5 5 14. What is the distance from the gas station to the farm? Express as a decimal rounded to the nearest tenth. 15. What is the distance from the gas station to the market? 16. What is the distance from the market to the farm? Express as a decimal rounded to the nearest tenth. -2 -4 M G -6