Midpoint andand Distance Midpoint Distance 18 2 in

Midpoint andand Distance Midpoint Distance 18. 2 in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Objectives Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints. Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 1 Find the coordinates of the midpoint of PQ with endpoints P(– 8, 3) and Q(– 2, 7). = (– 5, 5) Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 2 Find the coordinates of the midpoint of EF with endpoints E(– 2, 3) and F(5, – 3). Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 3: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y). Step 2 Use the Midpoint Formula: Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 3 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. 12 = 2 + x – 2 Simplify. Subtract. 10 = x Simplify. The coordinates of Y are (10, – 5). Holt Mc. Dougal Geometry 2=7+y – 7 – 5 = y

18. 2 Midpoint and Distance in the Coordinate Plane The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane. Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 4 Find FG and JK. Then determine whether FG JK. Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(– 4, 0), K(– 1, – 3) Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 4 Continued Step 2 Use the Distance Formula. Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 5: Finding Distances in the Coordinate Plane Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(– 2, – 5). Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 5 Continued Method 1 Use the Distance Formula. Substitute the values for the coordinates of D and E into the Distance Formula. Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Example 5 Continued Method 2 Use the Pythagorean Theorem. Count the units for sides a and b. a = 5 and b = 9. c 2 = a 2 + b 2 = 52 + 9 2 = 25 + 81 = 106 c = 10. 3 Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Lesson Quiz: Part I 1. Find the coordinates of the midpoint of MN with endpoints M(-2, 6) and N(8, 0). (3, 3) 2. K is the midpoint of HL. H has coordinates (1, – 7), and K has coordinates (9, 3). Find the coordinates of L. (17, 13) 3. Find the distance, to the nearest tenth, between S(6, 5) and T(– 3, – 4). 12. 7 4. The coordinates of the vertices of ∆ABC are A(2, 5), B(6, – 1), and C(– 4, – 2). Find the perimeter of ∆ABC, to the nearest tenth. 26. 5 Holt Mc. Dougal Geometry

18. 2 Midpoint and Distance in the Coordinate Plane Lesson Quiz: Part II 5. Find the lengths of AB and CD and determine whether they are congruent. Holt Mc. Dougal Geometry