document.write( "Question 94571This question is from textbook college algebra
\n" ); document.write( ": The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
\n" ); document.write( "(-3,1),(2,-1) and (6,9)\r
\n" ); document.write( "\n" ); document.write( "How is this problem solved? Please show all the steps clearly. Thank you. \n" ); document.write( "
Algebra.Com's Answer #68911 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
\n" ); document.write( "(-3,1),(2,-1) and (6,9)
\n" ); document.write( "How is this problem solved? Please show all the steps clearly. Thank you.
\n" ); document.write( "
\r\n" );
document.write( "First plot the points and draw the triangle so\r\n" );
document.write( "you can tell which one looks most like the \r\n" );
document.write( "vertex of a right angle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Well, from looking at that picture the angle \r\n" );
document.write( "at (2,-1) looks the most like a right angle.  \r\n" );
document.write( "So if this is really a right triangle, that \r\n" );
document.write( "would make the hypotenuse be the side connecting\r\n" );
document.write( "the points (-3,1) and (6,9), so we use the \r\n" );
document.write( "formula for the distance between two points to \r\n" );
document.write( "calculate the hypotenuse:\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(x2-x1)² + (y2-y1)²\r\n" );
document.write( "\r\n" );
document.write( "with (x1,y1) = (-3,1) and (x2,y2) = (6,9)  \r\n" );
document.write( "\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(6-(-3))² + (9-1)²\r\n" );
document.write( "     ___________\r\n" );
document.write( "D = Ö(6+3)² + 8²\r\n" );
document.write( "     _______\r\n" );
document.write( "D = Ö9² + 8²\r\n" );
document.write( "     _______ \r\n" );
document.write( "D = Ö81 + 64\r\n" );
document.write( "     ___\r\n" );
document.write( "D = Ö145 = hypotenuse\r\n" );
document.write( "\r\n" );
document.write( "Now we find the shorter leg, which connects \r\n" );
document.write( "the point (-3,1) to the point (2,-1)\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(x2-x1)² + (y2-y1)²\r\n" );
document.write( "\r\n" );
document.write( "with (x1,y1) = (-3,1) and (x2,y2) = (2,-1)  \r\n" );
document.write( "\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(2-(-3))² + (-1-1)²\r\n" );
document.write( "     ______________\r\n" );
document.write( "D = Ö(2+3)² + (-2)²\r\n" );
document.write( "     ______\r\n" );
document.write( "D = Ö5² + 4\r\n" );
document.write( "     ______ \r\n" );
document.write( "D = Ö25 + 4\r\n" );
document.write( "     __\r\n" );
document.write( "D = Ö29 = shorter leg\r\n" );
document.write( "\r\n" );
document.write( "Now we find the longer leg, from \r\n" );
document.write( "(-2,1) to (6,9)\r\n" );
document.write( "\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(x2-x1)² + (y2-y1)²\r\n" );
document.write( "\r\n" );
document.write( "with (x1,y1) = (-2,1) and (x2,y2) = (6,9)  \r\n" );
document.write( "Ö\r\n" );
document.write( "     __________________ \r\n" );
document.write( "D = Ö(6-2)² + (9-(-1))²\r\n" );
document.write( "     ___________\r\n" );
document.write( "D = Ö4² + (9+1)²\r\n" );
document.write( "     ________\r\n" );
document.write( "D = Ö16 + 10²\r\n" );
document.write( "     ________ \r\n" );
document.write( "D = Ö16 + 100\r\n" );
document.write( "     ___\r\n" );
document.write( "D = Ö116 = longer leg\r\n" );
document.write( "\r\n" );
document.write( "So we see if this Pythagorean equation holds:\r\n" );
document.write( "              ?\r\n" );
document.write( "(hypotenuse)² = (shorter leg)² + (longer leg)²\r\n" );
document.write( "        ___   ?   __      ___  \r\n" );
document.write( "      (Ö145)² = (Ö29)² + Ö116)²\r\n" );
document.write( "              ?  \r\n" );
document.write( "          145 = 29 + 116\r\n" );
document.write( "              Ö                \r\n" );
document.write( "          145 = 145\r\n" );
document.write( "\r\n" );
document.write( "So, yes, the Pythagorean equation holds so, \r\n" );
document.write( "it is a right triangle\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );