document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210512>Question 1210512</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Shown below is rectangle $EFGH$.  Its diagonals meet at $Y$.  Let $X$ be the foot if an altitude is dropped from $E$ to $\overline{FH}$.  If $FY = 24$ and $HX = 28$ and $EF = 10$, find the perimeter of rectangle $EFGH$.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853084 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=853084\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a challenging geometry problem that requires using properties of rectangles, the altitude theorem in right triangles, and the Pythagorean theorem.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's first establish the properties and known values. \r<BR>\n" );
document.write( "\n" );
document.write( "## 1. Establish Known Values and Properties\r<BR>\n" );
document.write( "\n" );
document.write( "| Property | Value | Reason |<BR>\n" );
document.write( "| :---: | :--- | :--- |<BR>\n" );
document.write( "| **$FY$** | 24 | Given |<BR>\n" );
document.write( "| **$EF$** | 10 | Given |<BR>\n" );
document.write( "| **$HX$** | 28 | Given |<BR>\n" );
document.write( "| **$EY$** | 24 | Diagonals of a rectangle are equal and bisect each other. (All four segments $EY, FY, GY, HY$ are equal). |<BR>\n" );
document.write( "| **$HY$** | 24 | Diagonals of a rectangle are equal and bisect each other. |<BR>\n" );
document.write( "| **$FH$** | 48 | $FH = FY + HY = 24 + 24$. $FH$ is the diagonal. |\r<BR>\n" );
document.write( "\n" );
document.write( "## 2. Find the Length of $FX$\r<BR>\n" );
document.write( "\n" );
document.write( "The segment $FX$ is part of the diagonal $FH$.<BR>\n" );
document.write( "$$FX = FH - HX$$<BR>\n" );
document.write( "$$FX = 48 - 28$$<BR>\n" );
document.write( "$$\mathbf{FX = 20}$$\r<BR>\n" );
document.write( "\n" );
document.write( "## 3. Find the Length of Altitude $EX$\r<BR>\n" );
document.write( "\n" );
document.write( "The triangle $\triangle EFH$ is a right triangle since $\angle FEH = 90^\circ$ (property of a rectangle).\r<BR>\n" );
document.write( "\n" );
document.write( "**Wait:** The angle $\angle HEF$ is $90^{\circ}$, but $\overline{EX}$ is the altitude dropped from $E$ to the hypotenuse $\overline{FH}$.<BR>\n" );
document.write( "* $EF$ is a leg of the right triangle $\triangle EFH$.\r<BR>\n" );
document.write( "\n" );
document.write( "We can use the **Altitude Theorem** (specifically, the Geometric Mean Theorem for a leg) on the right triangle $\triangle EFH$ with the altitude $\overline{EX}$.\r<BR>\n" );
document.write( "\n" );
document.write( "The relationship for the leg $\overline{EF}$ is:<BR>\n" );
document.write( "$$EF^2 = FX \cdot FH$$\r<BR>\n" );
document.write( "\n" );
document.write( "Let's check if the given values are consistent:<BR>\n" );
document.write( "$$10^2 = 20 \cdot 48$$<BR>\n" );
document.write( "$$100 = 960$$\r<BR>\n" );
document.write( "\n" );
document.write( "**The equation is FALSE.** $100 \neq 960$.\r<BR>\n" );
document.write( "\n" );
document.write( "This means the problem statement contains **contradictory geometric constraints**. It is impossible for a rectangle $EFGH$ to simultaneously have $FY=24$, $HX=28$, and $EF=10$.\r<BR>\n" );
document.write( "\n" );
document.write( "***\r<BR>\n" );
document.write( "\n" );
document.write( "## 💡 How to Proceed (Assuming a Typo in $EF$)\r<BR>\n" );
document.write( "\n" );
document.write( "Since a perimeter is requested, we must assume there was a typo and proceed by assuming **$EF^2 = FX \cdot FH$ is the correct theorem** and using the given values for $FY$, $HX$, and the properties of the rectangle to find $EF$.\r<BR>\n" );
document.write( "\n" );
document.write( "### Recalculating $EF$ (Assuming other values are correct)\r<BR>\n" );
document.write( "\n" );
document.write( "If $FY=24, HX=28$ are correct:<BR>\n" );
document.write( "* $FH = 48$<BR>\n" );
document.write( "* $FX = 20$<BR>\n" );
document.write( "* $EF^2 = FX \cdot FH = 20 \cdot 48 = 960$<BR>\n" );
document.write( "* $EF = \sqrt{960} = \sqrt{64 \cdot 15} = 8\sqrt{15}$\r<BR>\n" );
document.write( "\n" );
document.write( "### 4. Find the Length of $EH$ (The other side)\r<BR>\n" );
document.write( "\n" );
document.write( "We use the Pythagorean theorem on the right triangle $\triangle EFH$:<BR>\n" );
document.write( "$$EF^2 + EH^2 = FH^2$$\r<BR>\n" );
document.write( "\n" );
document.write( "Using $EF^2 = 960$ and $FH^2 = 48^2 = 2304$:<BR>\n" );
document.write( "$$960 + EH^2 = 2304$$<BR>\n" );
document.write( "$$EH^2 = 2304 - 960$$<BR>\n" );
document.write( "$$EH^2 = 1344$$<BR>\n" );
document.write( "$$EH = \sqrt{1344} = \sqrt{64 \cdot 21} = 8\sqrt{21}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### 5. Calculate the Perimeter\r<BR>\n" );
document.write( "\n" );
document.write( "The perimeter $P$ of rectangle $EFGH$ is $2(EF + EH)$.<BR>\n" );
document.write( "$$P = 2(8\sqrt{15} + 8\sqrt{21})$$<BR>\n" );
document.write( "$$P = 16(\sqrt{15} + \sqrt{21})$$\r<BR>\n" );
document.write( "\n" );
document.write( "***\r<BR>\n" );
document.write( "\n" );
document.write( "## Final Conclusion\r<BR>\n" );
document.write( "\n" );
document.write( "Due to the geometric contradiction ($100 \neq 960$), this problem has no solution as written.\r<BR>\n" );
document.write( "\n" );
document.write( "**If we assume the intent was to find the perimeter given $FY=24$ and $HX=28$ (forcing $EF=8\sqrt{15}$):**\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Perimeter} = \mathbf{16(\sqrt{15} + \sqrt{21})}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );