{"id":349,"date":"2015-09-18T22:41:38","date_gmt":"2015-09-18T22:41:38","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=349"},"modified":"2015-11-04T23:04:02","modified_gmt":"2015-11-04T23:04:02","slug":"using-the-midpoint-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-the-midpoint-formula\/","title":{"raw":"Using the Midpoint Formula","rendered":"Using the Midpoint Formula"},"content":{"raw":"When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].\r\n
[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\r\nA graphical view of a midpoint is shown in Figure 14. Notice that the line segments on either side of the midpoint are congruent.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"This Figure 14<\/b>[\/caption]\r\n\r\n
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Example 7: Finding the Midpoint of the Line Segment<\/h3>\r\nFind the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].\r\n\r\n<\/div>\r\n
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Solution<\/h3>\r\nUse the formula to find the midpoint of the line segment.\r\n
[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n
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Try It 2<\/h3>\r\nFind the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
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Example 8: Finding the Center of a Circle<\/h3>\r\nThe diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.\r\n\r\n<\/div>\r\n
\r\n

Solution<\/h3>\r\nThe center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.\r\n
[latex]\\begin{array}{c}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\r\n<\/div>","rendered":"

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula<\/strong>. Given the endpoints of a line segment, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[\/latex].<\/p>\n

[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/div>\n

A graphical view of a midpoint is shown in Figure 14. Notice that the line segments on either side of the midpoint are congruent.<\/p>\n

\"This<\/p>\n

Figure 14<\/b><\/p>\n<\/div>\n

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Example 7: Finding the Midpoint of the Line Segment<\/h3>\n

Find the midpoint of the line segment with the endpoints [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(9,5\\right)[\/latex].<\/p>\n<\/div>\n

\n

Solution<\/h3>\n

Use the formula to find the midpoint of the line segment.<\/p>\n

[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\hfill&=\\left(\\frac{7+9}{2},\\frac{-2+5}{2}\\right)\\hfill \\\\ \\hfill&=\\left(8,\\frac{3}{2}\\right)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n
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Try It 2<\/h3>\n

Find the midpoint of the line segment with endpoints [latex]\\left(-2,-1\\right)[\/latex] and [latex]\\left(-8,6\\right)[\/latex].<\/p>\n

Solution<\/a><\/p>\n<\/div>\n

\n

Example 8: Finding the Center of a Circle<\/h3>\n

The diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.<\/p>\n<\/div>\n

\n

Solution<\/h3>\n

The center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.<\/p>\n

[latex]\\begin{array}{c}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\n<\/div>\n\n\t\t\t
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