document.write( "Question 856091: a equilibrium triangle has a side length of 5 inches. what is the length of the altitude (height)
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Algebra.Com's Answer #515735 by Theo(13342)\"\" \"About 
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equilateral triangle has all sides equal to each other.
\n" ); document.write( "each side is therefore equal to 5.
\n" ); document.write( "if you draw an altitude from one of the vertices to the opposite side and perpendicular to that opposite side, you divide the triangle into 2 equal right triangles whose hypotenusse is equal to 5 and whose base is equal to 2.5 and whose altitude is equal to x because we don't know what it is.
\n" ); document.write( "by the pythagorean formula, we get:
\n" ); document.write( "a^2 + b^2 = c^2
\n" ); document.write( "in this triangle, c = 5, b = 2.5 and a = x
\n" ); document.write( "the formula becomes:
\n" ); document.write( "x^2 + 2.5^2 = 5^2
\n" ); document.write( "solve for x^2 in this equation to get:
\n" ); document.write( "x^2 = 5^2 - 2.5^2
\n" ); document.write( "this gets you x^2 = 18.75
\n" ); document.write( "take the square root of both sides of this equation to get:
\n" ); document.write( "x = sqrt(18.75) = 4.3301270129\r
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\n" ); document.write( "\n" ); document.write( "there is another way to find this.
\n" ); document.write( "all the angles of the equilateral triangle are equal to 60 degrees.
\n" ); document.write( "when you drop the perpendicular, you get 2 right triangles that have a 90 degree angle and a 60 degree angle and a 30 degree angle.
\n" ); document.write( "this is what is commonly known as a 30-60-90 degree triangle.
\n" ); document.write( "the sine of 60 degrees is equal to opposite side divided by hypotenuse.
\n" ); document.write( "the hypotenuse is equal to 5.
\n" ); document.write( "the opposite side is the altitude which is equal to x.
\n" ); document.write( "the sine of 60 degrees is equal to sqrt(3)/2
\n" ); document.write( "you get the formula:
\n" ); document.write( "sin(60) = x/5
\n" ); document.write( "multiply both sides of this equation by 5 to get:
\n" ); document.write( "5*sin(60) = x
\n" ); document.write( "replace sin(60) with sqrt(3)/2 to get:
\n" ); document.write( "5*sqrt(3)/2 = x
\n" ); document.write( "simplify to get:
\n" ); document.write( "2.5*sqrt(3) = x
\n" ); document.write( "if you use your calculator to multiply 2.5 * sqrt(3), you will get:
\n" ); document.write( "x = 2.5*sqrt(3) = 4.330127019
\n" ); document.write( "that's the same as we got above, so both methods get you the same answer.
\n" ); document.write( "this is good.
\n" ); document.write( "your answer is:
\n" ); document.write( "x = 2.5*sqrt(3) which can also be shown as:
\n" ); document.write( "x = 4.330127019
\n" ); document.write( "here's a picture of your equilateral triangle.
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