1.2.2 Midpoint and Distance in the Coordinate Plane Key Objectives • Develop and apply the formula for midpoint. • Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Key Terms • A coordinate plane is a plane that is divided into four regions by a horizontal line ( x -axis) and a vertical line ( y -axis). • In a right triangle, the two sides that form the right angle are the legs . • In a right triangle, the side across from the right angle that stretches from one leg to the other is the hypotenuse Formulas • Midpoint Formula + + M , 1 2 1 2 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are two points
x x y y 2 , 2
2 where ( x
• Distance Formula = − + − d x x y y ( ) ( ) , 2 1 2 2 1
1 , y 1 ) and ( x 2 , y 2 ) are two points
Theorems, Postulates, Corollaries, and Properties • Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Example 1 Finding the Coordinates of a Midpoint Given two points ( x 1 , y 1 ) and ( x 2 , y 2 ), the x -coordinate of the midpoint between those two
points is the average of the x -coordinates, ( x 1 + x 2 )/2. The y -coordinate of the midpoint between those two points is the average of the y -coordinates, ( y 1 + y 2 )/2.
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