document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169789>Question 1169789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a right pyramid ABCDE, on a square base ABCD, with AB = 8 cm, and height EO = 5 cm, what are the values of the following:\r<BR>\n" );
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document.write( "(a) angle EAB<BR>\n" );
document.write( "(b) angle β between a slant edge and the plane on the base.<BR>\n" );
document.write( "(c) angle θ between a slant face and the plane on the base. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851302 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=851302\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's break down this pyramid geometry problem.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "We have a right pyramid with a square base. We need to find specific angles related to the pyramid's edges and faces.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* Base ABCD is a square with side length AB = 8 cm.<BR>\n" );
document.write( "* Height of the pyramid EO = 5 cm (where O is the center of the square base).\r<BR>\n" );
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document.write( "**Diagram**\r<BR>\n" );
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document.write( "It's helpful to visualize the pyramid.\r<BR>\n" );
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document.write( "1.  Draw a square ABCD.<BR>\n" );
document.write( "2.  Mark the center of the square as point O.<BR>\n" );
document.write( "3.  Draw a line segment EO perpendicular to the square base, with EO = 5 cm.<BR>\n" );
document.write( "4.  Connect point E to each vertex of the square (A, B, C, D) to form the pyramid.\r<BR>\n" );
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document.write( "**Solutions**\r<BR>\n" );
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document.write( "**(a) Angle EAB**\r<BR>\n" );
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document.write( "1.  **Triangle EOA:** Triangle EOA is a right triangle, with ∠EOA = 90°.<BR>\n" );
document.write( "2.  **OA:** Since O is the center of the square, OA is half the length of the diagonal AC.<BR>\n" );
document.write( "    * AC = √(AB² + BC²) = √(8² + 8²) = √(128) = 8√2 cm.<BR>\n" );
document.write( "    * OA = (8√2) / 2 = 4√2 cm.<BR>\n" );
document.write( "3.  **Tangent:** We can use the tangent function to find angle EAB (let's call it α).<BR>\n" );
document.write( "    * tan(α) = EO / OA = 5 / (4√2)<BR>\n" );
document.write( "    * α = arctan(5 / (4√2)) ≈ arctan(5 / 5.6568) ≈ arctan(0.884)<BR>\n" );
document.write( "    * α ≈ 41.52°\r<BR>\n" );
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document.write( "Therefore, angle EAB ≈ 41.52°.\r<BR>\n" );
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document.write( "**(b) Angle β between a slant edge and the plane on the base.**\r<BR>\n" );
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document.write( "1.  **Slant Edge EB:** We need to find the angle between the slant edge EB and the base.<BR>\n" );
document.write( "2.  **Triangle EOB:** Triangle EOB is a right triangle, with ∠EOB = 90°.<BR>\n" );
document.write( "3.  **OB:** OB is half the length of the diagonal BD, which is equal to AC.<BR>\n" );
document.write( "    * OB = OA = 4√2 cm.<BR>\n" );
document.write( "4.  **Tangent:** We can use the tangent function to find angle EBO (β).<BR>\n" );
document.write( "    * tan(β) = EO / OB = 5 / (4√2)<BR>\n" );
document.write( "    * β = arctan(5 / (4√2)) ≈ 41.52°\r<BR>\n" );
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document.write( "Therefore, angle β ≈ 41.52°.\r<BR>\n" );
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document.write( "**(c) Angle θ between a slant face and the plane on the base.**\r<BR>\n" );
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document.write( "1.  **Slant Face EBC:** We need to find the angle between the slant face EBC and the base ABCD.<BR>\n" );
document.write( "2.  **Midpoint of BC:** Let M be the midpoint of BC.<BR>\n" );
document.write( "3.  **Triangle EOM:** Triangle EOM is a right triangle, with ∠EOM = 90°.<BR>\n" );
document.write( "4.  **OM:** OM is half the length of AB (or CD), so OM = 8 / 2 = 4 cm.<BR>\n" );
document.write( "5.  **Tangent:** We can use the tangent function to find angle EMO (θ).<BR>\n" );
document.write( "    * tan(θ) = EO / OM = 5 / 4 = 1.25<BR>\n" );
document.write( "    * θ = arctan(1.25) ≈ 51.34°\r<BR>\n" );
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document.write( "Therefore, angle θ ≈ 51.34°.\r<BR>\n" );
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document.write( "**Final Answers**\r<BR>\n" );
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document.write( "* **(a) Angle EAB ≈ 41.52°**<BR>\n" );
document.write( "* **(b) Angle β ≈ 41.52°**<BR>\n" );
document.write( "* **(c) Angle θ ≈ 51.34°**<BR>\n" );
document.write( " </SPAN>\n" );
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