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16 Sep 2014, 00:24
Kudos
Bookmarks
If a triangle has sides 7, 10, and \(x\), where \(x\) is an integer, how many different values can \(x\) take?
A. 8
B. 10
C. 13
D. 14
E. 16
A. 8
B. 10
C. 13
D. 14
E. 16
16 Sep 2014, 00:24
Kudos
Bookmarks
Official Solution:
If a triangle has sides 7, 10, and \(x\), where \(x\) is an integer, how many different values can \(x\) take?
A. 8
B. 10
C. 13
D. 14
E. 16
Important property: 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.
Using this property, we derive \(10 - 7 < x < 10 + 7\), which results in \(3 < x < 17\). Given that \(x\) is an integer, it can take 13 values, ranging from 4 to 16, inclusive.
Answer: C
If a triangle has sides 7, 10, and \(x\), where \(x\) is an integer, how many different values can \(x\) take?
A. 8
B. 10
C. 13
D. 14
E. 16
Important property: 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.
Using this property, we derive \(10 - 7 < x < 10 + 7\), which results in \(3 < x < 17\). Given that \(x\) is an integer, it can take 13 values, ranging from 4 to 16, inclusive.
Answer: C
Bunuel
For those who may be more visual in their approach to geometry problems, I am uploading a Google Doc I created to illustrate the rule. It may not be as fast as the official solution, but it can get the job done in a pinch. Happy studies.
- Andrew
*Edit (17 October 2021): I am editing the post to include an image of the document below. (You can still download the PDF at the bottom if you want.)
Attachment:
Screen Shot 2021-10-17 at 15.30.29.png [ 77.95 KiB | Viewed 5739 times ]
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