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16 Sep 2014, 00:38
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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
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
16 Sep 2014, 00:38
Kudos
Bookmarks
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
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
General Discussion
I think this is a high-quality question and I agree with explanation.
