Difference between revisions of "Alg6.2"

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For these pairs of points, find the midpoint, distance, slope, and equation of the line.<br><br>\n\n<math>(12,1),(4,0)\\,</math><br><br>\n\nTo find the midpoint, average the x coordinates and y coordinates. The midpoint is<br><br>\n\n<math>\\left(\\frac{12+4}{2},\\frac{1+0}{2}\\right) = \\left(8,\\frac{1}{2}\\right)\\,</math><br><br>\n\nTo find the (always zero or positive) distance, use the formula <math>d = +\\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\\,</math><br><br>\n\n<math>d = \\sqrt{(12-4)^2+(1-0)^2} = \\sqrt{(8)^2+1^2} = \\sqrt{64+1} = \\sqrt{5*15} = \\sqrt{5*3*5} = 5\\sqrt{3}\\,</math><br><br>\n\nTo find the slope, use the formula <math>m = \\frac{y_2-y_1}{x_2-x_1}\\,</math><br><br>\n\n<math>m = \\frac{0-1}{4-12} = \\frac{-1}{-8} = \\frac{1}{8}\\,</math><br><br>\n\nThe equations of the line are<br><br>\n\n<b>Form 1:</b> <math>y=mx+b\\,</math><br><br>\n\nPlug in one known point (say, <math>(4,0)\\,</math>) and the calculated slope.<br><br>\n\n<math>0 = \\frac{1}{8}\\cdot 4 + b\\,</math><br><br>\n\n<math>b = -\\frac{4}{8} = -\\frac{1}{2}\\,</math><br><br>\n\nNow plug <math>b</math> and <math>m</math> into the line equation:<br><br>\n\n*<math>y = \\frac{1}{8}x - \\frac{1}{2}\\,</math><br><br>\n\n<b>Form 2:</b> <math>(y-y_1) = m(x-x_1)\\,</math><br><br>\n\nPlug in one known point (say, <math>(12,1)\\,</math>) and the calculated slope.<br><br>\n\n<math>(y-1) = \\frac{1}{8}(x-12)\\,</math><br><br>\n\n<math>y = \\frac{1}{8}x - \\frac{12}{8} + 1 = \\frac{1}{8}x - \\frac{4}{8} \\,</math><br><br>\n\n*<math>y = \\frac{1}{8}x - \\frac{1}{2}\\,</math><br><br>
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For these pairs of points, find the midpoint, distance, slope, and equation of the line.<br><br>\n\n<math>(12,1),(4,0)\,</math><br><br>\n\nTo find the midpoint, average the x coordinates and y coordinates. The midpoint is<br><br>\n\n<math>\left(\frac{12+4}{2},\frac{1+0}{2}\right) = \left(8,\frac{1}{2}\right)\,</math><br><br>\n\nTo find the (always zero or positive) distance, use the formula <math>d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,</math><br><br>\n\n<math>d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5*15} = \sqrt{5*3*5} = 5\sqrt{3}\,</math><br><br>\n\nTo find the slope, use the formula <math>m = \frac{y_2-y_1}{x_2-x_1}\,</math><br><br>\n\n<math>m = \frac{0-1}{4-12} = \frac{-1}{-8} = \frac{1}{8}\,</math><br><br>\n\nThe equations of the line are<br><br>\n\n<b>Form 1:</b> <math>y=mx+b\,</math><br><br>\n\nPlug in one known point (say, <math>(4,0)\,</math>) and the calculated slope.<br><br>\n\n<math>0 = \frac{1}{8}\cdot 4 + b\,</math><br><br>\n\n<math>b = -\frac{4}{8} = -\frac{1}{2}\,</math><br><br>\n\nNow plug <math>b</math> and <math>m</math> into the line equation:<br><br>\n\n*<math>y = \frac{1}{8}x - \frac{1}{2}\,</math><br><br>\n\n<b>Form 2:</b> <math>(y-y_1) = m(x-x_1)\,</math><br><br>\n\nPlug in one known point (say, <math>(12,1)\,</math>) and the calculated slope.<br><br>\n\n<math>(y-1) = \frac{1}{8}(x-12)\,</math><br><br>\n\n<math>y = \frac{1}{8}x - \frac{12}{8} + 1 = \frac{1}{8}x - \frac{4}{8} \,</math><br><br>\n\n*<math>y = \frac{1}{8}x - \frac{1}{2}\,</math><br><br>
  
 
[[Main Page]] : [[Geometry]]
 
[[Main Page]] : [[Geometry]]

Revision as of 22:13, 7 November 2005

For these pairs of points, find the midpoint, distance, slope, and equation of the line.

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12,1),(4,0)\,}

\n\nTo find the midpoint, average the x coordinates and y coordinates. The midpoint is

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{12+4}{2},\frac{1+0}{2}\right) = \left(8,\frac{1}{2}\right)\,}

\n\nTo find the (always zero or positive) distance, use the formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,}

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5*15} = \sqrt{5*3*5} = 5\sqrt{3}\,}

\n\nTo find the slope, use the formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{y_2-y_1}{x_2-x_1}\,}

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{0-1}{4-12} = \frac{-1}{-8} = \frac{1}{8}\,}

\n\nThe equations of the line are

\n\nForm 1: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=mx+b\,}

\n\nPlug in one known point (say, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (4,0)\,} ) and the calculated slope.

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = \frac{1}{8}\cdot 4 + b\,}

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = -\frac{4}{8} = -\frac{1}{2}\,}

\n\nNow plug Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} into the line equation:

\n\n*Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{1}{8}x - \frac{1}{2}\,}

\n\nForm 2: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y-y_1) = m(x-x_1)\,}

\n\nPlug in one known point (say, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12,1)\,} ) and the calculated slope.

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y-1) = \frac{1}{8}(x-12)\,}

\n\nFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{1}{8}x - \frac{12}{8} + 1 = \frac{1}{8}x - \frac{4}{8} \,}

\n\n*Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{1}{8}x - \frac{1}{2}\,}

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