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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
50% (01:48) correct
50%
(02:04)
wrong
based on 190
sessions
History
Date
Time
Result
Not Attempted Yet
The angles in a triangle are \(x\), \(3x\), and \(5x\) degrees. If \(a\), \(b\) and \(c\) are the lengths of the sides opposite to angles \(x\), \(3x\), and \(5x\) respectively, then which of the following 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
M08-31
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
M08-31
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24 Jun 2020, 05:56
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VERBAL1
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 basically the same as I: if \(c=10\), \(a=6\) and \(b=2\) then \(c \gt a+b\), which can never be true. Thus even not 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, if interested why III is true: as the angles in a triangle are \(x\), \(3x\), and \(5x\) degrees then \(x+3x+5x=180\). Hence \(x=20\), \(3x=60\), and \(5x=100\). Next, if angle opposite side \(c\) were 90 degrees, then according to Pythagoras theorem \(c^2=a^2+b^2\), but since the angle opposite side \(c\) is more than 90 degrees (100) then side \(c\) is larger, hence \(c^2>a^2+b^2\).
Answer: E
24 Jun 2020, 08:27
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VERBAL1
The OA currently in the original post is "B", which is not correct (as Bunuel's solution demonstrates).
The sum of any two sides in a triangle always exceeds the third, so that alone lets us rule out I and II. For II, you can also recall that the ratio of a triangle's angles is almost never the same as the ratio of the lengths of its sides; that's the whole reason the subject of trigonometry is not trivially easy.
III is true because, with angles x, 3x and 5x, using the fact that these sum to 180, we find x = 20. So the largest angle is 100 degrees. When one angle in a triangle is precisely 90 degrees, a^2 + b^2 = c^2 is true, where c is the longest side. When you make the largest angle bigger than 90 degrees, that creates more space opposite that angle, making the opposite side longer. So in any triangle with an angle bigger than 90 degrees, c^2 > a^2 + b^2 is true, where c is the longest side.
But with these answer choices, it's not even necessary to confirm that III works, since I and II are both false by the triangle inequalities.
