document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210480>Question 1210480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the ratio of the area of the red region to the area of the yellow region.  Enter your answer as a fraction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852852 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=852852\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a trick question! The description is inconsistent.\r<BR>\n" );
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document.write( "* If the **red region is a triangle** and the **yellow region is the same triangle**, their areas must be **equal**.<BR>\n" );
document.write( "* If the **yellow region is the same triangle, half as big**, the yellow region cannot be the *same* triangle. It must be a **similar triangle** whose dimensions are scaled down.\r<BR>\n" );
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document.write( "Let's proceed with the most logical interpretation for a geometry/ratio problem: the yellow triangle is **similar** to the red triangle, with its linear dimensions scaled by a factor of $1/2$.\r<BR>\n" );
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document.write( "## 📐 Ratio of Areas for Similar Figures\r<BR>\n" );
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document.write( "When two geometric figures are **similar** (the same shape but different sizes), the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions (e.g., sides, heights).\r<BR>\n" );
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document.write( "Let:<BR>\n" );
document.write( "* $A_R$ be the area of the **red triangle**.<BR>\n" );
document.write( "* $A_Y$ be the area of the **yellow triangle**.<BR>\n" );
document.write( "* $s_R$ be a linear dimension (side length) of the red triangle.<BR>\n" );
document.write( "* $s_Y$ be the corresponding linear dimension of the yellow triangle.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 1: Determine the Ratio of Linear Dimensions\r<BR>\n" );
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document.write( "The problem states the yellow triangle is \"half as big.\" In geometry, \"half as big\" usually means the linear dimensions are halved:<BR>\n" );
document.write( "$$\frac{s_Y}{s_R} = \frac{1}{2}$$\r<BR>\n" );
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document.write( "### Step 2: Determine the Ratio of Areas\r<BR>\n" );
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document.write( "The ratio of the areas of two similar figures is the square of the ratio of their corresponding linear dimensions:<BR>\n" );
document.write( "$$\frac{A_Y}{A_R} = \left(\frac{s_Y}{s_R}\right)^2$$<BR>\n" );
document.write( "$$\frac{A_Y}{A_R} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$\r<BR>\n" );
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document.write( "### Step 3: Find the Required Ratio\r<BR>\n" );
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document.write( "The question asks for the ratio of the area of the **red region** to the area of the **yellow region** ($\frac{A_R}{A_Y}$):<BR>\n" );
document.write( "$$\frac{A_R}{A_Y} = \frac{1}{\frac{A_Y}{A_R}} = \frac{1}{\frac{1}{4}} = \mathbf{4}$$\r<BR>\n" );
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document.write( "The ratio of the area of the red region to the area of the yellow region is **$\frac{4}{1}$**.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "If you strictly follow the first part of the statement (\"The red region is a triangle, and the yellow region is the **same triangle**\"):<BR>\n" );
document.write( "* $A_R = A_Y$<BR>\n" );
document.write( "* Ratio $\frac{A_R}{A_Y} = \frac{1}{1}$\r<BR>\n" );
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document.write( "However, this makes the second part (\"half as big\") irrelevant or contradictory. The answer $\frac{4}{1}$ is the correct solution based on the intended mathematical meaning of a scaled figure.\r<BR>\n" );
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document.write( "Do you have another geometry problem you'd like to solve? </SPAN>\n" );
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