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Midpoint Intro If you measure a line segment that is 10 inches long and divide it into two 5-inch segments, you get two equal halves. The point at which you divide the line is called the midpoint. The midpoint of a line segment divides the segment into two congruent parts. Finding the midpoint of a line segment on a coordinate system is quite simple if the line is either horizontal or vertical; simply measure the length of the segment and divide by two. For example, E D A M M B Graph A Graph B In this example, the length of the vertical segment, AB, is 5 units. Therefore, the midpoint of AB is 2.5 units from either point. In this example, the length of the vertical segment, DE, is 4 units. Therefore, the midpoint of DE is 2 units from either point. Namely, the point M(-4 , -3.5). Namely, the point M(0 , 1). However, the process becomes a bit more challenging when trying to find the midpoint of a diagonal segment. Midpoint Formula Examining the two graphs more closely, we can determine how the midpoint is being calculated each time. This will help us in finding a general formula we can use for each scenario. Graph A shows a line segment, AB, whose coordinates are A(-4,1) and B(-4,-8); the midpoint of AB is M(-4,-3). If we think about the general concept of midpoint it is really finding the middle-point or average of two numbers. Following this thought process, we can see if finding the average of the x-values and the average of the y-values yields the midpoint: Midpoint of AB = = −4 + −4 1 + −8 and 2 2 −8 −7 and 2 2 = − 4 and − 3 .5 From this calculation, I can see finding the average of the x-values and y-values do indeed yield the midpoint. Try the same process with the information in graph B. x +x y +y The midpoint of a segment with endpoints (x 1 , y 1 ) and (x 2 , y 2 ) is 1 2 , 1 2 . Applications using the Midpoint Formula Example 1 Find the coordinates of the center of the rectangle. Solution You can find the center of any rectangle by finding the midpoint of either diagonal. Using the diagonal AC: 3+1 4 +1 , 2 2 4 5 = , 2 2 = (2 ,2.5 ) 2 2 Example 2 Find the midpoint of the hypotenuse for the right triangle, ΔBCD. Solution The hypotenuse of a right triangle is always the side opposite the right angle; therefore, the hypotenuse of ΔBCD is BD. Using BD: 1+3 3+1 , 2 2 4 4 = , 2 2 = (2,2) Example 3 10 cm A rectangle is 10 cm long and 7 cm wide. What is the area of the rhombus formed by connecting the midpoints of each side of the rectangle? Solution 7 cm A rhombus is comprised of two triangles. Draw a line which connects the midpoints of the lengths of the rectangle (in green). Next, we can draw a height for the triangles (in pink). The height of the triangle must be 5cm since the base connects the midpoints of the length of the rectangle. 1 2 Using our formula for finding the area of a triangle, bh , we can substitute our values: 1 (7)(5) = 17.5 . 2 Finally, since there are two triangles, we can double the area: 17.5(2) = 35 cm2.