Lesson A square divided into three equal areas by two parallel lines
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<H2>A square divided into three equal areas by two parallel lines</H2> <H3>Problem 1</H3>A square is divided into three equal areas by two parallel lines drawn from opposite vertices. Determine the area of the square if the distance between the two lines is 1 cm. <B>Solution</B> <pre> Make a plot following my description. Let ABCD be our square with the side length 'x', so A, B, C and D are its vertices. The lines divide our square into three equal areas, so the area of each part is {{{(1/3)x^2}}}. Draw the line AE from A to the side BC, so the intersection point E with BC divides side BC in proportion BE:CE = 2:1. In other words, BE = {{{(2/3)x}}}, XE = {{{(1/3)x}}}. Draw the line CF from the opposite vertex C to the side AD, so the intersection point F with AD divides side AD in proportion DF:AF = 2:1. In other words, DF = {{{(2/3)x}}}, AF = {{{(1/3)x}}}. So, now we have two right-angled triangles ABE and CDF of the area {{{(1/3)*x^2}}} each, and parallelogram AECF, whose area is also {{{(1/3)*x*2}}}, since it is the remaining area. So, now we have exactly the configuration described in the problem. We can easy find the lengths of intervals AE and CF as hypotenuses of triangles ABE and CFD AE = CF = {{{sqrt(x^2 + ((2/3)x)^2)}}} = {{{x*sqrt(1+4/9)}}} = {{{x*(sqrt(13)/3)}}}. Now the area of the parallelogram AECF is, from one hand side, {{{(1/3)*x^2}}}, and from other hand side it is the product of its base AE by the height, which is 1 cm. So, we can write this equation for the area of parallelogram AECF {{{(1/3)*x^2}}} = {{{x*(sqrt(13)/3)*1}}}. Cancel common factors, and you will get x = {{{sqrt(13))). Thus we found the side length of the square ABCD: it is {{{sqrt(13)}}} cm. Hence, the area of the square ABCD is {{{(sqrt(13))^2}}} = 13 cm^2. <<<---=== <U>ANSWER</U> </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/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</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>.
