document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151227>Question 1151227</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if an equilateral triangle has area equal to 10, what is its height?\r<BR>\n" );
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document.write( "A) 5 / sqrt(3)<BR>\n" );
document.write( "B) sqrt(10) * (root(4)(3))<BR>\n" );
document.write( "c) sqrt(40) / (root(4)(3)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772903 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=772903\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "h = height of equilateral triangle<BR>\n" );
document.write( "x = side length of equilateral triangle\r<BR>\n" );
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document.write( "Start with equilateral triangle ABC. Mark point D as the midpoint of one of the sides and draw a line to the opposing vertex. In this case, I marked the midpoint of AB and connected that to vertex C. This forms the height of the triangle. CD is perpendicular to AB. The base here is AB = x, half of which is x/2.<BR>\n" );
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document.write( "Focus on triangle CDB<BR>\n" );
document.write( "a = x/2 and b = h are the two legs<BR>\n" );
document.write( "c = x is the hypotenuse\r<BR>\n" );
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document.write( "Use the Pythagorean Theorem to solve for h<BR>\n" );
document.write( "a^2 + b^2 = c^2<BR>\n" );
document.write( "(x/2)^2 + h^2 = x^2<BR>\n" );
document.write( "(x^2)/4 + h^2 = x^2<BR>\n" );
document.write( "h^2 = x^2 - (x^2)/4<BR>\n" );
document.write( "h^2 = (4x^2)/4 - (x^2)/4<BR>\n" );
document.write( "h^2 = (4x^2-x^2)/4<BR>\n" );
document.write( "h^2 = (3x^2)/4<BR>\n" );
document.write( "h = sqrt( (3x^2)/4 )<BR>\n" );
document.write( "h = sqrt(3x^2)/sqrt(4)<BR>\n" );
document.write( "h = (sqrt(x^2)*sqrt(3))/sqrt(4)<BR>\n" );
document.write( "h = (x*sqrt(3))/2<BR>\n" );
document.write( "h = (x/2)*sqrt(3)\r<BR>\n" );
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document.write( "Plug this into the area of a triangle formula to get the area of an equilateral triangle formula<BR>\n" );
document.write( "A = (1/2)*b*h<BR>\n" );
document.write( "A = (1/2)*x*(x/2)*sqrt(3)<BR>\n" );
document.write( "A = (1/4)*x^2*sqrt(3)\r<BR>\n" );
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document.write( "Now plug in the given area A = 10 and solve for x.<BR>\n" );
document.write( "10 = (1/4)*x^2*sqrt(3)<BR>\n" );
document.write( "10*4 = x^2*sqrt(3)<BR>\n" );
document.write( "40 = x^2*sqrt(3)<BR>\n" );
document.write( "x^2*sqrt(3) = 40<BR>\n" );
document.write( "x^2 = 40/sqrt(3)<BR>\n" );
document.write( "x = sqrt(40/sqrt(3))<BR>\n" );
document.write( "x = sqrt(40)/sqrt(sqrt(3))<BR>\n" );
document.write( "x = sqrt(40)/(3^(1/4))<BR>\n" );
document.write( "x = sqrt(40)/root(4,3) <BR>\n" );
document.write( "x = sqrt(4*10)/root(4,3)<BR>\n" );
document.write( "x = sqrt(4)*sqrt(10)/root(4,3)<BR>\n" );
document.write( "x = 2*sqrt(10)/root(4,3)\r<BR>\n" );
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document.write( "Now use this to find h<BR>\n" );
document.write( "h = (x/2)*sqrt(3)<BR>\n" );
document.write( "h = (1/2)*sqrt(3)*x<BR>\n" );
document.write( "h = (1/2)*sqrt(3)*( 2*sqrt(10)/root(4,3) )<BR>\n" );
document.write( "h = sqrt(3)*sqrt(10)/root(4,3)<BR>\n" );
document.write( "h = sqrt(10)*sqrt(3)/root(4,3)<BR>\n" );
document.write( "h = sqrt(10)*(3^(1/2))/(3^(1/4))<BR>\n" );
document.write( "h = sqrt(10)*3^(1/2-1/4)<BR>\n" );
document.write( "h = sqrt(10)*3^(1/4)<BR>\n" );
document.write( "h = sqrt(10)*root(4,3)\r<BR>\n" );
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document.write( "<font color=red size=4>Answer: B) sqrt(10)*root(4,3)</font>\r<BR>\n" );
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document.write( "Note: root(4,3) refers to the 4th root of 3. So root(4,3) can be written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C3%29 ALIGN=MIDDLE  ALT=\"root%284%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In terms of exponents, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C3%29+=+3%5E%281%2F4%29%5E%22%22 ALIGN=MIDDLE  ALT=\"root%284%2C3%29+=+3%5E%281%2F4%29%5E%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here is an alternate way to get the answer\r<BR>\n" );
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document.write( "x = half the side length of the equilateral triangle (eg: DB = x)<BR>\n" );
document.write( "h = height of equilateral triangle<BR>\n" );
document.write( "a^2 + b^2 = c^2 ... pythagorean theorem<BR>\n" );
document.write( "x^2 + h^2 = (2x)^2<BR>\n" );
document.write( "x^2 + h^2 = 4x^2<BR>\n" );
document.write( "h^2 = 3x^2<BR>\n" );
document.write( "h = sqrt(3x^2)<BR>\n" );
document.write( "h = x*sqrt(3)<BR>\n" );
document.write( "x = h/sqrt(3)<BR>\n" );
document.write( "2x = 2h/sqrt(3) is the full base of the equilateral triangle\r<BR>\n" );
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document.write( "A = 0.5*base*height<BR>\n" );
document.write( "A = 0.5*2x*h<BR>\n" );
document.write( "A = 0.5*(2h/sqrt(3))*h<BR>\n" );
document.write( "A = h^2*3^(-1/2)<BR>\n" );
document.write( "10 = h^2*3^(-1/2)<BR>\n" );
document.write( "10*3^(1/2) = h^2<BR>\n" );
document.write( "h = (10*3^(1/2))^(1/2)<BR>\n" );
document.write( "h = (10)^(1/2)*(3^(1/2))^(1/2)<BR>\n" );
document.write( "h = sqrt(10)*(3^(1/2))^(1/2)<BR>\n" );
document.write( "h = sqrt(10)*3^(1/4)<BR>\n" );
document.write( "h = <font color=red>sqrt(10)*root(4,3)</font>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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