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01 Jul 2019, 12:52
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
51% (02:50) correct
49%
(02:40)
wrong
based on 258
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The measures of the interior angles in a polygon are consecutive odd integers. The largest angle measures 153 degrees. How many sides does this polygon have?
A) 8
B) 9
C) 10
D) 11
E) 12
A) 8
B) 9
C) 10
D) 11
E) 12
01 Jul 2019, 12:55
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dabaobao
Solution: Using Sequence
Sum of Interior angles = (n-2) * 180
Max Angle = 153; Min Angle = 153 - 2(n-1)
Since we have an evenly spaced set, sum of interior angles = ((a1 + an)/2)*n
((153 - 2(n-1) + 153)/2)*n = (n-2) * 180
(153 -n + 1)*n = 180n - 360 => 153n -n^2 +n = 180n - 360
=> n^2 + 26n - 360 = 0 => (n + 36) (n - 10) = 0 => n = 10
ANSWER: C
Veritas Prep Official Solution
Angles of the polygon: 153, 151, 149, 147, 145, 143, 141, … , (153 – 2(n-1))
The average of these angles must be equal to the measure of each interior angle of a regular polygon with n sides since the sum of all angles is the same in both the cases.
Measure of each interior angle of n sided regular polygon = Sum of all angles / n = (n−2)∗180n
Using the options:
Measure of each interior angle of 8 sided regular polygon = 180*6/8 = 135 degrees
Measure of each interior angle of 9 sided regular polygon = 180*7/9 = 140 degrees
Measure of each interior angle of 10 sided regular polygon = 180*8/10 = 144 degrees
Measure of each interior angle of 11 sided regular polygon = 180*9/11 = 147 degrees apprx
and so on…
Notice that the average of the given angles can be 144 if there are 10 angles.
The average cannot be higher than 144 i.e. 147 since that will give us only 7 sides (153, 151, 149, 147, 145, 143, 141 – the average is 147 is this case). But the regular polygon with interior angle measure of 147 has 11 sides. Similarly, the average cannot be less than 144 i.e. 140 either because that will give us many more sides than the required 9.
Hence, the polygon must have 10 sides.
Answer (C).
Similar questions to practice:
the-measures-of-the-interior-angles-in-a-polygon-are-127388.html
the-measures-of-the-interior-angles-in-a-polygon-151005.html
General Discussion
03 Jul 2019, 19:39
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The sum of the interior angles of a ‘n’ sided polygon = (n-2) * 180 degrees.
Since the interior angles are consecutive odd integers, they represent a set of equally spaced values. For a set of equally spaced values,
Mean = \(\frac{First element + Last element}{2}\).
The largest angle = 153 degrees. So, the smallest angle = 153 – 2(n-1). Therefore,
Mean angle = \(\frac{153 + 153 – 2n + 2}{2}\) = 153 – n + 1.
But, Mean angle = \(\frac{Sum of angles}{No. of angles}\).
Therefore, 153 – n + 1 = \(\frac{180 n – 360}{n}\).
Beyond this stage, we can start plugging in the options, starting with option C.
If n = 10, 153 – 10 + 1 = \(\frac{180 * 10 – 360}{10}\) i.e. 144 = 144.
The equation is satisfied when we plug in n = 10. Therefore, the correct answer option is C.
Hope this helps!
Since the interior angles are consecutive odd integers, they represent a set of equally spaced values. For a set of equally spaced values,
Mean = \(\frac{First element + Last element}{2}\).
The largest angle = 153 degrees. So, the smallest angle = 153 – 2(n-1). Therefore,
Mean angle = \(\frac{153 + 153 – 2n + 2}{2}\) = 153 – n + 1.
But, Mean angle = \(\frac{Sum of angles}{No. of angles}\).
Therefore, 153 – n + 1 = \(\frac{180 n – 360}{n}\).
Beyond this stage, we can start plugging in the options, starting with option C.
If n = 10, 153 – 10 + 1 = \(\frac{180 * 10 – 360}{10}\) i.e. 144 = 144.
The equation is satisfied when we plug in n = 10. Therefore, the correct answer option is C.
Hope this helps!
