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1.1 Sets of Real Numbers and the Cartesian Coordinate Plane 13 M Σ 2 , 2 + xy + y 0101 x . M = Equation 1.2.The Midpoint Formula: The midpoint M of the line segment connecting P (x 0 , y 0 ) and Q (x 1 , y 1 ) is: Q ( x 1 , y 1 ) P ( x 0 , y 0 ) If we think of reaching M by going ‘halfway over’ and ‘halfway up’ we get the following formula. If we let d denote the distance between P and Q , we leave it as Exercise 36 to show that the distance between P and M is d/ 2 which is the same as the distance between M and Q . This suffices to show that Equation 1.2 gives the coordinates of the midpoint. Example 1.1.6. Find the midpoint of the line segment connecting P ( 2 , 3) and Q (1 , 3). Solution. M = . x 0 + x 1 , y 0 + y 1 Σ = . ( 2 ) + 1 , 3 + ( 3) Σ = . 1 , 0 Σ The midpoint is . 1 , 0 Σ . 2 2 2 2 = . 1 , 0 Σ We close with a more abstract application of the Midpoint Formula. We will revisit the following example in Exercise 72 in Section 2.1 . Example 1.1.7. If a ƒ = b , prove that the line y = x equally divides the line segment with endpoints ( a, b ) and ( b, a ). Solution. To prove the claim, we use Equation 1.2 to find the midpoint