\text{Ratio} = \frac{A_{\text{new}}}{A_{\text{original}}} = \frac{\frac{2.25\sqrt{3}}{4}s^2}{\frac{\sqrt{3}}{4}s^2} = \frac{2.25\sqrt{3}/4}{\sqrt{3}/4} = 2.25
["# Understanding the Ratio: ( \frac{A_{\ ext{new}}}{A_{\ ext{original}}} = 2.25 ) — A Step-by-Step Explanation", "When dealing with geometric shapes — especially triangles — ratios of areas play a crucial role in simplifying complex problems. One powerful geometric ratio commonly encountered is:", "[\n\frac{A_{\ ext{new}}}{A_{\ ext{original}}} = \frac{\frac{2.25\sqrt{3}}{4}s^2}{\frac{\sqrt{3}}{4}s^2} = \frac{2.25\sqrt{3}/4}{\sqrt{3}/4} = 2.25\n]", "But what does this ratio truly mean? How is it derived, and why does it equal 2.25? Let’s break it down clearly and thoroughly.", "---", "## What Is This Ratio?", "This ratio calculates how the area of a new geometric figure relates to the area of an original figure, specifically when scaling or transforming a shape — usually equilateral triangles — by a consistent factor.", "Given:", "- Original area: ( A_{\ ext{original}} = \frac{\sqrt{3}}{4}s^2 )\n- New area: ( A_{\ ext{new}} = \frac{2.25\sqrt{3}}{4}s^2 )", "We compute the ratio:", "[\n\frac{A_{\ ext{new}}}{A_{\ ext{original}}} = \frac{\frac{2.25\sqrt{3}}{4}s^2}{\frac{\sqrt{3}}{4}s^2}\n]", "Note: since both numerator and denominator contain the same ( s^2 ) and ( \frac{1}{4} ) factor, these cancel out.", "---", "## Step-by-Step Simplification", "Start with:", "[\n\frac{\frac{2.25\sqrt{3}}{4}s^2}{\frac{\sqrt{3}}{4}s^2}\n]", "Cancel ( s^2 ), ( \frac{1}{4} ), and ( \sqrt{3} ) (since both numerator and denominator share these):", "[\n\frac{2.25}{\sqrt{3}} \div \frac{1}{\sqrt{3}} = \frac{2.25}{\sqrt{3}} \ imes \frac{\sqrt{3}}{1} = 2.25 \cdot \frac{\sqrt{3}}{\sqrt{3}} = 2.25 \cdot 1 = 2.25\n]", "Thus,\n[\n\frac{A_{\ ext{new}}}{A_{\ ext{original}}} = 2.25\n]", "---", "## What Does a Ratio of 2.25 Mean?", "A ratio of ( 2.25 ) indicates that the new shape’s area is 2.25 times larger than the original area. This commonly arises when scaling a figure by a linear factor. In geometric similarity:", "[\n\frac{A_{\ ext{new}}}{A_{\ ext{original}}} = \left(\frac{\ ext{new side length}}{\ ext{original side length}}\right)^2\n]", "Since ( 2.25 = \frac{9}{4} = (1.5)^2 ), the side length was scaled by ( 1.5 ) (or 150%) — a 50% increase.", "---", "## Practical Applications", "This calculation applies in various mathematical and real-world scenarios:", "- Architecture: Comparing scaled models to full-size structures.\n- Engineering: Analyzing changes in surface area during design modifications.\n- Math education: Reinforcing similarity and proportion concepts in geometry.", "Understanding such ratios helps solve area scaling problems swiftly and precisely.", "---", "## Summary", "The ratio\n[\n\frac{A_{\ ext{new}}}{A_{\ ext{original}}} = \frac{2.25\sqrt{3}/4\cdot s^2}{\sqrt{3}/4\cdot s^2} = 2.25\n]\nshows that doubling and scaling an equilateral triangle’s side by 1.5 increases its area by 125% more — exactly 2.25 times larger.", "Mastering these ratios empowers deeper geometric reasoning and simplifies complex transformations.", "---", "Key takeaway: When comparing area ratios derived from scaling side lengths, always simplify constants carefully — even when radicals are present — to arrive at clean numerical results like 2.25."]
