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M12-06

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Bunuel
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M12-06 [#permalink] New post   16 Sep 2014, 00:46
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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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C
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Bunuel
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Re M12-06 [#permalink] New post   16 Sep 2014, 00:46
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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°.



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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sgrover18
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Re: M12-06 [#permalink] 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.

Regards,
Shradha
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Re: M12-06 [#permalink] 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] 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] New post   02 Dec 2017, 04:26
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ranaazad wrote:
Doesn't shortest side mean shortest leg and the longest side mean the bigger leg of a right triangle? :?

Nunuel, please clarify


No. The longest side in a right triangle is the hypotenuse.
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Re: M12-06 [#permalink] 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 [#permalink] New post   07 Apr 2021, 10:22
Given the longest side is 2x and the angle that corresponds to this side is 90 degrees, can you set 90 equal to 2x and solve for x?
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Re: M12-06 [#permalink] New post   07 Apr 2021, 10:29
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rsidawi wrote:
Given the longest side is 2x and the angle that corresponds to this side is 90 degrees, can you set 90 equal to 2x and solve for x?


No. 2x would correspond to the length of the hypotenuse while 90° is an angle measure. You can not equate length to the angle measure.
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Re: M12-06 [#permalink]
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