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# M12-06

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Math Expert
Joined: 02 Sep 2009
Posts: 52971
M12-06  [#permalink]

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15 Sep 2014, 23:46
00:00

Difficulty:

25% (medium)

Question Stats:

76% (00:34) correct 24% (00:38) wrong based on 90 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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Posts: 52971
Re M12-06  [#permalink]

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15 Sep 2014, 23:46
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°.

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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Re: M12-06  [#permalink]

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16 Nov 2016, 12:34
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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Joined: 24 Mar 2017
Posts: 1
Re: M12-06  [#permalink]

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22 Jul 2017, 20:21
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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02 Dec 2017, 02:02
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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02 Dec 2017, 03:26
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]

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02 Dec 2017, 22:36
I believe the only trick in this question is realizing that the longest side means the hypotenuse and not the LONG side
Re: M12-06   [#permalink] 02 Dec 2017, 22:36
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# M12-06

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