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13 Nov 2018, 01:19
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Difficulty:
65%
(hard)
Question Stats:
39% (00:55) correct
61%
(00:57)
wrong
based on 269
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In the figure above, triangle MNO is equilateral. What is the area of triangle MNO ?
(1) NP has a length of \(5\sqrt{3}\).
(2) MN has a length of 10.
Attachment:
2018-11-13_1215.png [ 33.02 KiB | Viewed 6298 times ]
13 Nov 2018, 01:30
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Bunuel
(D) is our answer.
There are two different approaches to solving this question.
A Precise approach relies on knowledge of geometric rules:
(1) NP is a leg in a 30-60-90 triangle NPO meaning we can calculate the hypotenuse (which is the side of the triangle). Then we have the height and base of a triangle and can calculate its area.
(2) a useful equation to remember is that the area of an equilateral triangle is \(\frac{a^2\sqrt{3}}{4}\), where 'a' is the length of a side.
(Remember 1-2-3-4 i.e. length - square - sqrt(3) - div by 4)
Then both are sufficient.
A second approach, which does not rely on knowledge of geometric rules is Alternative.
An equilateral triangle is a regular polygon and there is only ONE way to draw a regular polygon.
That is - once you know the length of just one of the sides/heights/chords/etc. you know all of the shape's properties. (Try it out - you can SEE this)
So both (1) and (2) are sufficient.
13 Nov 2018, 02:02
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The area of equilateral triangle is = \(s^2 *\sqrt{3}/4\)
where the height is\(s * \sqrt{3}/2\)
30:60:90
\(s:s\sqrt{3}:2s\)
we know the height is the side that corresponds to the 60 angle so \(s\sqrt{3} = 5\sqrt{3}\)
From that we can find the side s = 5 and accordingly find the area of the triangle.
Statement 1 is sufficient.
For statement 2
We are given a side of the equilateral triangle so using the equation we can find the area.
Sufficient.
Answer choice D
where the height is\(s * \sqrt{3}/2\)
30:60:90
\(s:s\sqrt{3}:2s\)
we know the height is the side that corresponds to the 60 angle so \(s\sqrt{3} = 5\sqrt{3}\)
From that we can find the side s = 5 and accordingly find the area of the triangle.
Statement 1 is sufficient.
For statement 2
We are given a side of the equilateral triangle so using the equation we can find the area.
Sufficient.
Answer choice D
