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M12-06

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M12-06  [#permalink]

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New post 16 Sep 2014, 00:46
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75% (00:33) correct 25% (00:38) wrong based on 88 sessions

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If in a right triangle, the ratio of the shortest side to the longest is \(\frac{1}{2}\), what is the smallest angle in this triangle?

A. 15 degrees
B. 20 degrees
C. 30 degrees
D. 45 degrees
E. 60 degrees

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Re M12-06  [#permalink]

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New post 16 Sep 2014, 00:46
Official Solution:

If in a right triangle, the ratio of the shortest side to the longest is \(\frac{1}{2}\), what is the smallest angle in this triangle?

A. 15 degrees
B. 20 degrees
C. 30 degrees
D. 45 degrees
E. 60 degrees


Since the ratio of hypotenuse to the smallest leg in a right triangle is \(\frac{1}{2}\), then it means that we have 30°-60°-90° right triangle. So, the smallest angle is 30°.

A right triangle where the angles are 30°, 60°, and 90°.

Image

This is one of the 'standard' triangles you should be able to recognize on sight. A fact you should commit to memory is: The sides are always in the ratio \(1 : \sqrt{3}: 2\).

Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).


Answer: C
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Collection of Questions:
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Re: M12-06  [#permalink]

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New post 16 Nov 2016, 13:34
In case, one doesn't catch the ratio:

Let the three sides be a, b & c. where a<b<c or a=b <c

& \(a^2 + b^2 = c^2\) since the given triangle is a right triangle.

Now as per the question : \(\frac{a}{c}\) = \(\frac{1}{2}\)
Square both the sides: \(\frac{a^2}{c^2}\) = \(\frac{1}{4}\)
putting the value of \(c^2\) : \(\frac{a^2}{a^2+b^2}\) = \(\frac{1}{4}\)

Cross multiply: 4\(a^2\) = \(a^2 + b^2\)
Further simplify: \(b^2\) = 3\(a^2\)
=> \(\frac{a}{b}\)= \(\frac{1}{\sqrt{3}}\)

by now we should know that the triangle is a 1: \(\sqrt{3}\):2 = 30:60:90

Therefore the smallest angle is 30 degrees.

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Shradha
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Re: M12-06  [#permalink]

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New post 22 Jul 2017, 21:21
I solved this a little differently, I would like to get some feedback.

Since its a right triangle we know one side is 90 degrees

so

180=90+x(second biggest angle)+.5x(small angle)

X= 60
Smalles angle is 30
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Re: M12-06  [#permalink]

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New post 02 Dec 2017, 03:02
Doesn't shortest side mean shortest leg and the longest side mean the bigger leg of a right triangle? :?

Nunuel, please clarify
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Re: M12-06  [#permalink]

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New post 02 Dec 2017, 04:26
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Re: M12-06  [#permalink]

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New post 02 Dec 2017, 23:36
I believe the only trick in this question is realizing that the longest side means the hypotenuse and not the LONG side
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Re: M12-06 &nbs [#permalink] 02 Dec 2017, 23:36
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