document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209518>Question 1209518</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (36) PQRS is a rhombus, W is the midpoint of PQ, T is the midpoint of PS and V is the intersection point of QT and SW. What fraction of rhombus PQRS is quadrilateral PWVT?\r<BR>\n" );
document.write( "\n" );
document.write( "Link to diagram: https://ibb.co/PsrxgPwG </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849140 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849140\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Area of Rhombus:**\r<BR>\n" );
document.write( "\n" );
document.write( "Let the side length of the rhombus be 's'.  The area of a rhombus can be calculated using its diagonals, but since we're dealing with midpoints and ratios, a simpler approach is to consider it as four congruent triangles.  We'll denote the area of rhombus PQRS as [PQRS].\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Area of Triangles:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Triangle PQT:** Since W and T are midpoints, PW = WQ = s/2 and PT = TS = s/2.  Triangle PQT has half the base and half the height of triangle PQS (which is half of the rhombus). Therefore, [PQT] = (1/2) * (s/2) * (s/2) * sin(∠P) = (1/8) * s² * sin(∠P) = (1/4) * [PQS] = (1/8) * [PQRS].\r<BR>\n" );
document.write( "\n" );
document.write( "* **Triangles PSW and QRS:** By the same logic, [PSW] = (1/8) * [PQRS].\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Area of Triangle PVW:**\r<BR>\n" );
document.write( "\n" );
document.write( "We know that the medians of a triangle divide it into six equal areas. In triangle PQS, QT and SW are medians. Therefore, the area of triangle PVW is 1/6 the area of triangle PQS.<BR>\n" );
document.write( "[PVW] = (1/6) * [PQS] = (1/6) * (1/2) [PQRS] = (1/12) * [PQRS]\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Area of Quadrilateral PWVT:**\r<BR>\n" );
document.write( "\n" );
document.write( "The area of quadrilateral PWVT can be found by subtracting the areas of triangles PVW, PWT, and QWV from the area of triangle PQT.\r<BR>\n" );
document.write( "\n" );
document.write( "*Note that PWT and QWV are also 1/6 of the area of the relevant larger triangle, which is 1/4 of the area of the rhombus.*<BR>\n" );
document.write( "[PWT] = [QWV] = (1/6) * [PQT] = (1/6) * (1/8) [PQRS] = (1/48) [PQRS]\r<BR>\n" );
document.write( "\n" );
document.write( "[PWVT] = [PQT] - [PVW] - [PWT]<BR>\n" );
document.write( "[PWVT] = (1/8)[PQRS] - (1/12)[PQRS] - (1/48)[PQRS]<BR>\n" );
document.write( "[PWVT] = ((6-4-1)/48) * [PQRS]<BR>\n" );
document.write( "[PWVT] = (1/48) * [PQRS]\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Fraction of Rhombus:**\r<BR>\n" );
document.write( "\n" );
document.write( "The area of quadrilateral PWVT is (1/48) of the area of rhombus PQRS.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the fraction of rhombus PQRS that is quadrilateral PWVT is **1/48**.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );