{"id":9348,"date":"2020-12-04T12:40:22","date_gmt":"2020-12-04T07:10:22","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=9348"},"modified":"2020-12-04T17:45:37","modified_gmt":"2020-12-04T12:15:37","slug":"midpoint-line-segment","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/midpoint-line-segment\/","title":{"rendered":"Midpoint of a Line Segment"},"content":{"rendered":"
The point halfway between the endpoints of a line segment is called the midpoint<\/strong>. A midpoint divides a line segment into two equal segments. Method 1:<\/strong> Find the midpoints \\(\\overline { AB }\\) and \\(\\overline { CD }\\).<\/strong> Method 2:<\/strong> This concept of finding the average of the coordinates can be written as a formula: Consider this “tricky” midpoint problem: Other Methods of Solution:<\/strong> Utilizing the concept of slope and congruent triangles:<\/strong> Find the slope between points C and M. This slope has a run of 2 units to the left and a rise of 4 units up. By repeating this slope from point M (move 2 units to the left and 4 units up), you will arrive at the other endpoint. Midpoint of a Line Segment The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal segments. By definition, a midpoint of a line segment is the point on that line segment that divides the segment two congruent segments. In Coordinate Geometry, there […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[3440],"yoast_head":"\n
\nBy definition, a midpoint of a line segment<\/strong> is the point on that line segment that divides the segment two congruent segments.
\nIn Coordinate Geometry, there are several ways to determine the midpoint of a line segment.<\/p>\n
\nIf the line segments are vertical or horizontal, you may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.<\/p>\n
\nAB is 8 (by counting). The midpoint is 4 units from either endpoint. On the graph, this point is (1,4).
\nCD is 3 (by counting). The midpoint is 1.5 units from either endpoint. On the graph, this point is (2,1.5)<\/p>\n
\nIf the line segments are diagonally positioned, more thought must be paid to the solution. When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates.<\/p>\n
\nNOTE:<\/strong> The Midpoint Formula works for all line segments: vertical, horizontal or diagonal.
\n<\/p>\n
\nM is the midpoint of \\(\\overline { CD }\\). The coordinates M(-1,1) and C(1,-3) are given.\u00a0<\/strong>Find the coordinates of point D.<\/strong>
\nFirst, visualize the situation. This will give you an idea of approximately where point D will be located. When you find your answer, be sure it matches with your visualization of where the point should be located.
\n
\n<\/p>\n
\nVerbalizing the algebraic solution:<\/strong>
\nSome students like to do these “tricky” problems by just examining the coordinates and asking themselves the following questions:
\n“My midpoint’s x-coordinate is -1. What is -1 half of? (Answer -2)
\nWhat do I add to my endpoint’s x-coordinate of +1 to get -2? (Answer -3)
\nThis answer must be the x-coordinate of the other endpoint.”
\nThese students are simply verbalizing the algebraic solution.
\n(They use the same process for the y-coordinate.)<\/p>\n
\nA line segment is part of a straight line whose slope (rise\/run) remains the same no matter where it is measured. Some students like to look at the rise and run values of the x and y coordinates and utilize these values to find the missing endpoint.<\/p>\n
\nBy using this slope approach, you are creating two congruent right triangles whose legs are the same lengths. Consequently, their hypotenuses are also the same lengths and DM = MC making M the midpoint of \\(\\overline { CD }\\).<\/p>\n","protected":false},"excerpt":{"rendered":"