Transcript 2 + - Quia
Lesson 1-3 Distance and Midpoints Lesson Outline Five-Minute Check Then & Now and Objectives Vocabulary Key Concept Examples Lesson Checkpoints Summary and Homework Then and Now You graphed points on the coordinate plane. • Find the distance between two points • Find the midpoint of a (line) segment Objectives • Find the distance between two points • Find the midpoint of a (line) segment Vocabulary • Distance – the length of a segment connecting two points • Midpoint – the point halfway between the endpoints of a segment • Segment Bisector – any segment, line or plane that intersects the segment at its midpoint • Irrational number – any real number that cannot be expressed as a ratio a / b, where a and b are integers, with b non-zero Key Concept • In Geometry we always have positive distances (hence the use of the absolute value signs) • Most people just count it off on a number line Example 1 Use the number line to find QR. The coordinates of Q and R are –6 and –3. QR = | –6 – (–3) | = | –3 | or 3 Answer: 3 Distance Formula Simplify. Key Concept • The distance formula is an application of the Pythagorean Theorem: PQ² = (x2 - x1)² + (y2 - y1)² with the change in x the horizontal line and the change in y the vertical line. Example 2 Find the distance between E(–4, 1) and F(3, –1). Method 1 Pythagorean Theorem Use the gridlines to form a triangle so you can use the Pythagorean Theorem. Simplify. Take the square root of each side. Example 2 cont Method 2 Distance Formula Distance Formula Simplify. Simplify. Answer: The distance from E to F is units. You can use a calculator to find that is approximately 7.28. Key Concept • The midpoint is halfway between the endpoints of a segment. • The midpoint is analogous to the average or the mean of two numbers. Example 3 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. x1 + x2 M = -----------2 Midpoint Formula 2.5 + 10 M = -----------2 x1 = 2.5, x2 = 10 M = 6.25 Simplify. Answer: The midpoint of the couch back is 6.25 ft from the wall. Key Concept • The midpoint must lie on the line connecting the two points (at the halfway point). Example 4 Find the coordinates of M, the midpoint of for G(8, –6) and H(–14, 12). Let G be and H be , . y x Answer: (–3, 3) Example 5 Find the coordinates of D if E(–6, 4) is the midpoint of and F has coordinates (–5, –3). Let F be in the Midpoint Formula. Write two equations to find the coordinates of D. Example 5 cont Solve each equation. Multiply each side by 2. Add 5 to each side. Multiply each side by 2. Add 3 to each side. Answer: The coordinates of D are (–7, 11). Example 6 What is the measure of PR if Q is the midpoint of PR? Plan Because Q is the midpoint, you know that QR = (1/2) PR. Solve QR = (1/2)(PR) Definition of midpoint 6 – 3x = (1/2)(14x + 2) QR = 6 – 3x, PR = 14x + 2 6 – 3x = 7x + 1 Distributive Property Example 6 cont 5 – 3x = 7x Subtract 1 from each side. 5 = 10x Add 3x to each side ½=x Divide each side by 10 Now substitute ½ for x in the expression for PR PR = 14x + 2 Original measure PR = 7 + 2 or 9 Simplify Answer: The measure of PR is 9 units Distance and Midpoints Review Concept Midpoint Nr line Formula Examples (a + b) 2 (2 + 8) 2 [x2+x1] , [y2+y1] 2 2 Coord Plane Distance Nr line Coord Plane D=|a–b| (x2-x1)2 + (y2-y1)2 D= =5 7 + 1 , 4 + 2 = (4, 3) 2 2 D = | 2 – 8| = 6 D = (7-1)2 + (4-2)2 = 40 Y (7,4) a 1 2 D b 3 4 5 6 7 8 9 (1,2) ∆y ∆x X Lesson Checkpoints Summary & Homework • Summary: – Distances • can be determined on a number line • can be determined on the coordinate plane by using the Distance Formula or by Pythagorean Theorem – The midpoint of a segment is the point halfway between the segment’s endpoints • like an average of the endpoints • Homework: – pg 30-3: 13-15, 27-29, 41, 42, 47, 48