Difference between revisions of "1956 AHSME Problems/Problem 30"
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+ | == Problem 30== | ||
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+ | If the altitude of an equilateral triangle is <math>\sqrt {6}</math>, then the area is: | ||
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+ | <math> \textbf{(A)}\ 2\sqrt{2}\qquad\textbf{(B)}\ 2\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 6\sqrt{2}\qquad\textbf{(E)}\ 12 </math> | ||
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==Solution== | ==Solution== | ||
Drawing the altitude, we see that is opposite the <math>60^{\circ}</math> angle in a <math>30-60-90</math> triangle. Thus, we find that the shortest leg of that triangle is <math>\sqrt{\frac{6}{3}} = \sqrt{2}</math>, and the side of the equilateral triangle is then <math>2\sqrt{2}</math>. Thus, the area is | Drawing the altitude, we see that is opposite the <math>60^{\circ}</math> angle in a <math>30-60-90</math> triangle. Thus, we find that the shortest leg of that triangle is <math>\sqrt{\frac{6}{3}} = \sqrt{2}</math>, and the side of the equilateral triangle is then <math>2\sqrt{2}</math>. Thus, the area is | ||
− | <cmath>\frac{(2\sqrt 2)^2 \cdot \sqrt{3}}{4} = \frac{8 \sqrt{3}}{4} = 2 \sqrt {3 | + | <cmath>\frac{(2\sqrt 2)^2 \cdot \sqrt{3}}{4}</cmath> <cmath>= \frac{8 \sqrt{3}}{4}</cmath> <cmath>= 2 \sqrt {3}</cmath> |
<math>\boxed{B}</math> | <math>\boxed{B}</math> | ||
~JustinLee2017 | ~JustinLee2017 | ||
+ | |||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME 50p box|year=1956|num-b=29|num-a=31}} | ||
+ | {{MAA Notice}} |
Latest revision as of 22:30, 12 February 2021
Problem 30
If the altitude of an equilateral triangle is , then the area is:
Solution
Drawing the altitude, we see that is opposite the angle in a
triangle. Thus, we find that the shortest leg of that triangle is
, and the side of the equilateral triangle is then
. Thus, the area is
~JustinLee2017
See Also
1956 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 31 | |
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All AHSME Problems and Solutions |
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