document.write( "Question 844469: A rectangle is inscribed fully inside an equilateral triangle. The triangle has a side length of 10. The rectangle shares one side with a triangle side (but the rectangle side is shorter than the triangle side. The rectangle's other two vertices each touch another side of the triangle.\r
\n" ); document.write( "\n" ); document.write( "I already know the entire triangle has an area of 25 radical 3. I would like to find the area of the rectangle. Or the area of the triangle excluding the rectangle. \n" ); document.write( "

Algebra.Com's Answer #508786 by josgarithmetic(39617) $\"\"$ $\"About$

You can put this solution on YOUR website!
This may be difficult to follow without showing a diagram, figure, but think this way.\r
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\n" ); document.write( "\n" ); document.write( "Cutting the triangle in half from one vertex to middle of the opposite side, you have a 30-60-90 triangle which you can fit onto a cartesian coordinate system. Putting the right angle at the origin, the triangle has vertices (0,0), (0,10), and (5,0). Equation for the line joining (0,10) and (5,0) is $\"highlight_green%28y=-2x%2B10%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Pick any (x,y) point on the line, and HALF of the rectangle in your description is x horizontally and y vertically. The area of your WHOLE rectangle will be $\"highlight_green%282%2Ax%2Ay%29\"$ . That is two variables, and $\"0%3Cx%3C5\"$ and $\"0%3Cy%3C10\"$ . Reminder, x here is equivalent from middle of the side of your equilateral to the endpoint of the rectangle's side. \r
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\n" ); document.write( "\n" ); document.write( "We here have a formula for y, directly from the line in the graph I described. Area is $\"2xy=2x%28-2x%2B10%29=highlight%28-4x%5E2%2B20x%29\"$ .
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\n" ); document.write( "The actual rectangle's length horizontally will be 2x, and the tallness will be y. Pick your x, as long as it is $\"0%3Cx%3C5\"$ and use the linear equation to find the corresponding y value. The area for the rectangle will be $\"highlight%2820x-4x%5E2%29\"$ . \n" ); document.write( "

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