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Distance and Midpoint Formulas: Examples and Applications

Learn how to use the distance and midpoint formulas on a number line and coordinate plane. Practice finding distances, midpoints, and coordinates of endpoints. Discover real-world applications of these concepts.

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Distance and Midpoint Formulas: Examples and Applications

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  1. Concept 13 Distance and Midpoints

  2. Five-Minute Check (over Lesson 1–2) CCSS Then/Now New Vocabulary Key Concept: Distance Formula (on Number Line) Example 1: Find Distance on a Number Line Key Concept: Distance Formula (in Coordinate Plane) Example 2: Find Distance on Coordinate Plane Key Concept: Midpoint Formula (on Number Line) Example 3: Real-World Example: Find Midpoint on Number Line Key Concept: Midpoint Formula (in Coordinate Plane) Example 4: Find Midpoint in Coordinate Plane Example 5: Find the Coordinates of an Endpoint Example 6: Use Algebra to Find Measures Lesson Menu

  3. A.x = 2, AB = 8 B.x = 1, AB = 5 C. D.x = –2, AB = –4 What is the value of x and AB if B is between A and C, AB = 3x + 2, BC = 7, and AC = 8x – 1? 5-Minute Check 1

  4. If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, what is the value of x and MN? A.x = 1, MN = 0 B.x = 2, MN = 1 C.x = 3, MN = 2 D.x = 4, MN = 3 5-Minute Check 2

  5. . A. B. C. D. in. . in. Find RT. 5-Minute Check 3

  6. What segment is congruent to MN? A.MQ B.QN C.NQ D. no congruent segments 5-Minute Check 4

  7. What segment is congruent to NQ? A.MN B.NM C.QM D. no congruent segments 5-Minute Check 5

  8. A. 5 B. 6 C. 14 D. 18 5-Minute Check 6

  9. Content Standards G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Mathematical Practices 2 Reason abstractly and quantitatively. 7 Look for and make use of structure. CCSS

  10. You graphed points on the coordinate plane. • Find the distance between two points. • Find the midpoint of a segment. Then/Now

  11. Concept

  12. Find Distance on a Number Line 1. Use the number line to find QR. The coordinates of Q and R are –6 and –3. QR = | –6 – (–3) | Distance Formula = | –3 | or 3 Simplify. Answer: 3 Example 1

  13. 2. Use the number line to find AX. A. 2 B. 8 C. –2 D. –8 Example 1

  14. Concept

  15. (x1, y1) = (–4, 1) and (x2, y2) = (3, –1) Find Distance on a Coordinate Plane 3. Find the distance between E(–4, 1) and F(3, –1). Example 2

  16. A.4 B. C. D. 4. Find the distance between A(–3, 4) and M(1, 2). Example 2

  17. Midpoint – a point that is in the middle of 2 points, halfway between each point. PQ = QR PR = 2(PQ) PR = 2(QR)

  18. Concept

  19. Find Midpoint on a Number Line 5. Marco places a couch so that its end is perpendicular and 2.5 feet away from the wall. The couch is 90” wide. How far is the midpoint of the couch back from the wall in feet? First we must convert 90 inches to 7.5 feet. The coordinates of the endpoints of the couch are 2.5 and 10. Let M be the midpoint of the couch. Midpoint Formula x1 = 2.5, x2 = 10 Simplify. Answer: The midpoint of couch back is 6.25 feet from the wall. Example 3

  20. 6. The length of a drag racing strip is mile long. How many feet from the finish line isthe midpoint of the racing strip? A. 330 ft B. 660 ft C. 990 ft D. 1320 ft Example 3

  21. Concept

  22. Find Midpoint in Coordinate Plane 7. Answer: (–3, 3) Example 4

  23. 8. A. (–10, –6) B. (–5, –3) C. (6, 12) D. (–6, –12) Example 4

  24. (x2, y2) = (–5, –3) Find the Coordinates of an Endpoint 9. Let D be (x1, y1) and F be (x2, y2) in the Midpoint Formula. Write two equations to find the coordinates of D. Example 5

  25. Midpoint Formula Midpoint Formula Find the Coordinates of an Endpoint Answer: The coordinates of D are (–7, 11). Example 5

  26. 10. Find the coordinates ofRifN (8, –3) is the midpointofRSandShas coordinates (–1, 5). A. (3.5, 1) B. (–10, 13) C. (15, –1) D. (17, –11) Example 5

  27. Use Algebra to Find Measures 11. Example 6

  28. 12. A. 1 B. 10 C. 5 D. 3 Example 6

  29. Segment Bisector – a segment or line that goes through the middle of a segment, can create a midpoint.

  30. End of the Lesson

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