<ol class="num-list"><li><div class="list-item-paragraph">The ratio of the areas of two similar figures is the square of the ratio of their corresponding lengths, i.e. \(\frac{A_1}{A_2} = \left( \frac{l_1}{l_2} \right)^2\) where&nbsp;\(A_1\) and \(l_1\) are the area and the length of the first figure respectively, and \(A_2\) and \(l_2\) are the area and the length of the second figure respectively.</div></li><li><div class="list-item-paragraph"> If the triangles are <u>not similar</u>, then check if they have a <u>common height</u>.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1767503163989-el9159hlgrb.png" alt="List item image" class="aligncenter custom-width" style="max-width: min(200px, 100%); height: auto;" /></div><div class="list-item-paragraph">For triangles with <u>common height</u>, the ratio of the areas of the two triangles is <u>equal</u> to the ratio of the lengths of the bases of the two triangles.</div></li></ol>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: center;"><p style="text-align: right">\(\frac{A_1}{A_2} = \frac{\frac{1}{2} \times b_1 \times h}{\frac{1}{2} \times b_2 \times h} = \frac{b_1}{b_2}\)</p></div><div class="question-column" style="flex: 1; align-self: center;"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 200px;"></colgroup><tbody><tr style="height: 43px;"><td rowspan="2" style="height: 43px; min-height: 43px; text-align: left; --cell-justify: flex-start"><p>Triangles with common height are not similar.</p></td></tr><tr style="height: 40px;"></tr></tbody></table></div></div></div></div>
<p style="margin-left: 3rem">where&nbsp;\(b_1\) and \(A_1\) is the base and area of \(\Delta ABD\) respectively, </p>
<p style="margin-left: 3rem">and \(b_2\) and \(A_2\) is the base and area of \(\Delta BCD\) respectively.</p>
<ol class="num-list" start="3"><li><div class="list-item-paragraph">The ratio of the volumes of two similar solids is the cube of the ratio of their corresponding lengths, i.e. \(\frac{V_1}{V_2} = \left( \frac{l_1}{l_2} \right)^3\) where&nbsp;\(V_1\) and \(l_1\) are the volume and the length of the first solid respectively, and \(V_2\) and \(l_2\) are the volume and the length of the second solid respectively.</div></li><li><div class="list-item-paragraph">Masses is proportional to volumes, i.e. \(\frac{m_1}{m_2} = \frac{V_1}{V_2}\) and \(\frac{m_1}{m_2} = \left( \frac{l_1}{l_2} \right)^3\) where&nbsp;\(m_1\) is the mass of the first solid, and \(m_2\) is the mass of the second solid.</div></li></ol>

The ratio of the areas of two similar figures is the square of the ratio of their corresponding lengths, i.e. \(\frac{A_1}{A_2} = \left( \frac{l_1}{l_2} \right)^2\) where \(A_1\) and \(l_1\) are the area and the length of the first figure respectively, and \(A_2\) and \(l_2\) are the area and the length of the second figure respectively.

 If the triangles are <u>not similar</u>, then check if they have a <u>common height</u>.

For triangles with <u>common height</u>, the ratio of the areas of the two triangles is <u>equal</u> to the ratio of the lengths of the bases of the two triangles.

where \(b_1\) and \(A_1\) is the base and area of \(\Delta ABD\) respectively, 

and \(b_2\) and \(A_2\) is the base and area of \(\Delta BCD\) respectively.

The ratio of the volumes of two similar solids is the cube of the ratio of their corresponding lengths, i.e. \(\frac{V_1}{V_2} = \left( \frac{l_1}{l_2} \right)^3\) where \(V_1\) and \(l_1\) are the volume and the length of the first solid respectively, and \(V_2\) and \(l_2\) are the volume and the length of the second solid respectively.

Masses is proportional to volumes, i.e. \(\frac{m_1}{m_2} = \frac{V_1}{V_2}\) and \(\frac{m_1}{m_2} = \left( \frac{l_1}{l_2} \right)^3\) where \(m_1\) is the mass of the first solid, and \(m_2\) is the mass of the second solid.

Area and Volume of Similar Figures and Solids

Area and Volume of Similar Figures and Solids

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