document.write( "Question 456884: The sides of a length of a triangle are x, x+4, and 20, where 20 is the longest side. For which range of values is x an acute triangle \n" ); document.write( "

Algebra.Com's Answer #313550 by robertb(5830) $\"\"$ $\"About$

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For the triangle to be acute, $\"a%5E2+%2B+b%5E2+%3E+c%5E2\"$ , where c is the longest side of the triangle. Note that it is given that the longest side has length 20.\r
\n" ); document.write( "\n" ); document.write( "Hence, we must have $\"x%5E2+%2B+%28x%2B4%29%5E2+%3E+20%5E2\"$ , or , after simplifying, $\"x%5E2+%2B+4x+-+192+%3E0\"$ , or (x+16)(x - 12) >0. Since x must have positive values only, x > 12.\r
\n" ); document.write( "\n" ); document.write( "From the triangle inequality, we get the relation x +(x+4) > 20, or x > 8. Also, since 20 is the longest side, we must have x + 4 < 20, or x < 16. Hence from the initial conditions, we must have 8 < x < 16.\r
\n" ); document.write( "\n" ); document.write( "Intersect the preceding interval with the interval x > 12.\r
\n" ); document.write( "\n" ); document.write( "Therefore, for the triangle to be acute, we must have 12 < x < 16. \n" ); document.write( "

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