document.write( "Question 1186931: An isosceles triangle ABC, in which AB = BC = 6√2 and AC = 12 is folded along the altitude BD so that planes ABD and BDC form a right dihedral angle. Find the angle between side AB and its new
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Algebra.Com's Answer #848638 by yurtman(42)\"\" \"About 
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**1. Find the Height (Altitude) of the Isosceles Triangle**\r
\n" ); document.write( "\n" ); document.write( "* **Use the Pythagorean Theorem:**
\n" ); document.write( " * In right triangle ABD (where D is the midpoint of AC):
\n" ); document.write( " * AB² = AD² + BD²
\n" ); document.write( " * (6√2)² = AD² + 6²
\n" ); document.write( " * 72 = AD² + 36
\n" ); document.write( " * AD² = 36
\n" ); document.write( " * AD = 6\r
\n" ); document.write( "\n" ); document.write( "**2. Visualize the Folding**\r
\n" ); document.write( "\n" ); document.write( "* When triangle ABC is folded along altitude BD, side AB rotates around point B.
\n" ); document.write( "* Imagine the original position of AB and its new position after folding. This creates a dihedral angle (the angle between the two planes).\r
\n" ); document.write( "\n" ); document.write( "**3. Determine the Angle Between AB and its New Position**\r
\n" ); document.write( "\n" ); document.write( "* The angle between AB and its new position is twice the angle between AB and the plane BDC.
\n" ); document.write( "* Let's call this angle θ.\r
\n" ); document.write( "\n" ); document.write( "* **Consider right triangle ABD:**
\n" ); document.write( " * tan(θ) = AD / BD = 6 / 6 = 1
\n" ); document.write( " * θ = arctan(1) = 45°\r
\n" ); document.write( "\n" ); document.write( "* **Angle between AB and its new position:** 2 * θ = 2 * 45° = 90°\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the angle between side AB and its new position after folding is 90 degrees.**
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