M08-31

Official Solution:


In a triangle, the angles measure \(x\), \(3x\), and \(5x\) degrees. If the opposite sides of these angles have lengths \(a\), \(b\), and \(c\), respectively, which of the following statements must be true ?

I. \(c \gt a+b\)

II. \(c:a:b=10:6:2\)

III. \(c^2 \gt a^2+b^2\)


A. I and III only
B. II and III only
C. I only
D. II only
E. III only


According to the relationship of the sides of a triangle: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Thus I and II can never be true: one side (\(c\)) cannot be larger than the sum of the other two sides (\(a\) and \(b\)). Note that II is essentially the same as I: if \(c=10\), \(a=6\), and \(b=2\), then \(c \gt a+b\), which can never be true. Thus, without considering the angles, we can see that only answer choice E (III only) is left (all other options are out because each of them has either I or II in them).

Now, let's explore why III is true: given that the angles in a triangle are \(x\), \(3x\), and \(5x\) degrees, we have \(x+3x+5x=180\). Therefore, \(x=20\), \(3x=60\), and \(5x=100\). If the angle opposite side \(c\) were 90 degrees, then according to the Pythagorean theorem, \(c^2=a^2+b^2\). However, since the angle opposite side \(c\) is more than 90 degrees (100), side \(c\) must be larger. Thus, \(c^2 \gt a^2+b^2\).


Answer: E
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

Easy and very simple solution, thanks team!

I think this is a high-quality question.

I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question.

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.

Hi Bunuel,

I understand the solution. I am little confused regarding proportion of sides. I thought proportion of sides should be equal to proportion of angles of triangle. Can you clarify?

Thanks in advance.

In a triangle, the ratio of the sides is not equal to the ratio of the angles. The only triangle where the ratio of the sides is equal to the ratio of the angles, is the equilateral triangle. In an equilateral triangle, the side lengths are in ratio 1:1:1, as are the angle measures.
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