A Seemingly Impossible Problem
<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-a-seemingly-impossible-problem" target="_blank">A Seemingly Impossible Problem Exercise</a></p><p>The reason why we memorize the ratios of the side lengths in $30$-$60$-$90$ and $45$-$45$-$90$ triangles is because doing so can lead to us solving <strong><span style="color:#8e44ad;">Seemingly Impossible Problems</span></strong>. </p>
<p>For example, how would we calculate the area of the triangle below?</p>
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<p>The first thing we do is draw a from point $B$ to point $D$ so that it creates a right angle at point $D$.</p>
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<p>This allows us to then fill in all of the angles throughout the now two triangles. </p>
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<p>Notice how the original $105$° $B$ angle is now split into a $45$° angle and a $60$° angle. </p>
<p>Using what we have memorized about $30$-$60$-$90$ and $45$-$45$-$90$ triangles, we can fill the other sides in (in red). </p>
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<p>Finally, we know the "base" of the triangle, $6 + 6\sqrt{3}$, and the height, $6$, so we can calculate the area:</p>
<p>$$\frac{(6+6\sqrt{3})6}{2} = 18 + 18\sqrt{3}$$</p>