Locating Points and Midpoints LESSON 1–3 Locating Points and Midpoints

Five-Minute Check (over Lesson 1–2) TEKS Then/Now New Vocabulary Key Concept: Midpoint Formula (on Number Line) Example 1: Real-World Example: Find Midpoint on Number Line Key Concept: Midpoint Formula (in Coordinate Plane) Example 2: Find Midpoint in Coordinate Plane Example 3: Finding Missing Coordinates Example 4: Find Missing Measures Example 5: Locating a Point at Fractional Distances Example 6: Fractional Distances on a Coordinate Plane Example 7: Locating a Point Given a Ratio Lesson Menu

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

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

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

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

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

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

Mathematical Processes G.1(A), G.1(G) Targeted TEKS G.2(A) Determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint. G.2(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. Also addresses G.5(B). Mathematical Processes G.1(A), G.1(G) TEKS

You found the distance between two points on a line segment. Find the midpoint of a segment. Locate a point on a segment given a fractional distance from one endpoint. Then/Now

midpoint segment bisector Vocabulary

Concept

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 1

Answer: The midpoint of the couch back is 6.25 feet from the wall. Find Midpoint on a Number Line Simplify. Answer: The midpoint of the couch back is 6.25 feet from the wall. Example 1

DRAG RACING The length of a drag racing strip is. mile long DRAG RACING The length of a drag racing strip is mile long. How many feet from the finish line is the midpoint of the racing strip? A. 330 ft B. 660 ft C. 990 ft D. 1320 ft Example 1

Concept

Concept

Find Midpoint in Coordinate Plane Answer: (–3, 3) Example 2

A. (–10, –6) B. (–5, –3) C. (6, 12) D. (–6, –12) Example 2

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

Answer: The coordinates of D are (–7, 11). Find Missing Coordinates Midpoint Formula Midpoint Formula Answer: The coordinates of D are (–7, 11). Example 3

Find the coordinates of R if N (8, –3) is the midpoint of RS and S has coordinates (–1, 5). B. (–10, 13) C. (15, –1) D. (17, –11) Example 3

Use Algebra to Find Measures You 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. Example 4

Because Q is the midpoint, you know that Use Algebra to Find Measures Because Q is the midpoint, you know that Use this equation and the algebraic measures to find a value for x. Subtract 1 from each side. Example 4

Use Algebra to Find Measures Original measure Example 4

QR = 6 – 3x Original Measure Use Algebra to Find Measures QR = 6 – 3x Original Measure Example 4

Use Algebra to Find Measures Multiply. Simplify. Example 4

A. 1 B. 10 C. 5 D. 3 Example 4

Locating a Point at Fractional Distances   BD = |x2 – x1| Distance Formula = |-2 - -6| x1 = -6, x2 = -2 = |4| or 4 Subtract. The distance between B and D is 4. Example 5

Locating a Point at Fractional Distances   Answer: -5 Example 5

  A. -7 B. 0 C. 2 D. -1 Example 5

Fractional Distances on a Coordinate Plane   Example 6

  Example 6

  Answer: P (–1.3, -1) Example 6

  A. (2, 1.8) B. (-2, -1.8) C. (-1.8, -2) D. (1.8, 2) Example 6

Locating a Point Given a Ratio    

Locating a Point Given a Ratio   Answer: F (–2, 4.4)

  A. (3, -0.4) B. (3, 0.4) C. (-0.4, 8.9) D. (-0.4, 3.1)

Locating Points and Midpoints LESSON 1–3 Locating Points and Midpoints