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Geometry Essentials: Distance and Midpoint Formulas for Coordinate Plane

Learn how to find distances on number lines, coordinates of endpoints, and midpoints in the coordinate plane with real-world examples and algebraic measures.

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Geometry Essentials: Distance and Midpoint Formulas for Coordinate Plane

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 1–2) 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 B C D 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 AB = 8x – 1? 5-Minute Check 1

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

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

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

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

  9. You graphed points on the coordinate plane. (Lesson 0–2) • Find the distance between two points. • Find the midpoint of a segment. Then/Now

  10. distance • midpoint • segment bisector Vocabulary

  11. Concept

  12. Find Distance on a Number Line 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. A B C D 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 Find the distance between E(–4, 1) and F(3, –1). Example 2

  16. Find Distance on a Coordinate Plane CheckGraph the ordered pairs and check by using the Pythagorean Theorem. Example 2

  17. . Find Distance on a Coordinate Plane Example 2

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

  19. Concept

  20. Find Midpoint on a Number Line DECORATING 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 Example 3

  21. Find Midpoint on a Number Line Simplify. Answer: The midpoint of the couch back is 6.25 feet from the wall. Example 3

  22. A B C D DRAG RACING 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

  23. Concept

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

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

  26. (x2, y2) = Find the Coordinates of an Endpoint 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

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

  28. A B C D 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

  29. UnderstandYou know that Q is the midpoint of PR, and the figure gives algebraic measures for QR and PR. You are asked to find the measure of PR. Use Algebra to Find Measures Example 6

  30. PlanBecause Q is the midpoint, you know that Subtract 1 from each side. Use Algebra to Find Measures Use this equation and the algebraic measures to find a value for x. Solve Example 6

  31. Original measure Use Algebra to Find Measures Example 6

  32. Check Use Algebra to Find Measures QR = 6 – 3x Original Measure Example 6

  33. Multiply. Simplify. Use Algebra to Find Measures Example 6

  34. A B C D A. 1 B. 3 C. 5 D. 10 Example 6

  35. End of the Lesson

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