Lesson Find the angle between sides of folded triangle
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<H2>Find the angle between sides of folded triangle</H2> <H3>Problem 1</H3>An isosceles triangle ABC, in which AB = BC = {{{6*sqrt(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 position. <B>Solution</B> <pre> In the isosceles triangle ABC AB^2 + BC^2 = {{{(6*sqrt(2))^2}}} + {{{(6*sqrt(2))^2}}} = 36*2 + 36*2 = 72 + 72 = 144 and AC^2 = {{{12^2}}} = 144. It means that triangle ABC is a right-angled isosceles triangle with the right angle at B, hypotenuse BC and the legs AB and BC. Now imagine that the triangle ABC is folded along the altitude BD so the planes ABD and BDC form a right dihedral angle. Let plane BDC remains in its original position and vertex A takes new position A' in space. Then the segments A'D and CD form the right angle. Consider triangle A'DC. It is a right-angled triangle with the right angle D and the legs A'D and CD. These legs have equal length A'D = CD = 12/2 = 6, A'D= AD and AD and CD are equal halves of the base AC of the triangle ABC. Therefore, the hypotenuse A'C of the triangle ADC has the length {{{sqrt(AD^2 + CD^2)}}} = {{{sqrt(6^2+6^2)}}} = {{{6*sqrt(2)}}}. Thus the triangle A'BC (after folding) has congruent sides A'B, A'C and BC: the length of each of these sides is {{{6*sqrt(2)}}}. Thus the triangle A'BC is an equilateral triangle and all its angles have the measure of 60°. In particular, the angle A'BC is 60°. The angle between AB and its new position A'B is the same 60°. The proof is the same. </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
