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08 Dec 2018, 10:16
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C
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Difficulty:
95%
(hard)
Question Stats:
31% (02:01) correct
69%
(02:56)
wrong
based on 71
sessions
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In the given figure, if AS = 10 cm, SN = 5 cm and TN = 8 cm, which of the following could be the positive difference between the maximum and minimum value of AT, if all the sides shown in the figure are positive integers?
A) 16
B) 18
C) 20
D) 21
E) 23
Attachment:
Ques fig.jpg
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10 Dec 2018, 07:40
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Official Solution:
Given:
AS = 10 cm
SN = 5 cm
TN = 8 cm
Find:
What could be the value of AT?
Since we have been given the lengths of the triangle and we do not have any information on the angles of the triangles, we will have to use the side property of the triangle to solve this question.
In triangle ASN –AS = 10 cm, SN = 5 cm
We know that sum of two sides > third side and we also know that difference of two sides < third side, thus, we can write –
10-5 < AN < 10+5
5 < AN < 15
Thus the value of AN can be {6,7,8,9,10,11,12,13,14}
Find the maximum value of AT. In triangle ATN, we can write –
AT < AN + TN
AT < 14 + 8
AT < 22
Thus, we can say that the maximum value of AT is 21 units.
Now, let us find the minimum value of AT
In triangle ATN, we can write –
TN ~ AN < AT
From the possible value of AN - {6,7,8,9,10,11,12,13,14}, if we take AN = 8, we will get AT > 8 – 8
AT > 0
Thus, the minimum value of AT will be 1 unit.
Therefore, the positive difference between the maximum and minimum value of AT could be = ( 21 – 1) = 20 units
The correct answer is Option C.
General Discussion
08 Dec 2018, 11:20
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No figure appears.
Now figure has finally appeared!
The rule is that the sum of the lengths of any two sides of a triangle is greater the length of the third. Thus considering triangle ANS first, we have:
5 + AN > 10 => AN > 5. Also AN < 15, hence 5 < AN < 15. Since only integral values are allowed, AN = {6, 7, 8,...,14}.
Using only the extremal values of AN in triangle ATN, we get:
Use AN = 6 to get minimum value of AT: AT + 6 > 8 => AT > 2; therefore minimum value = 3.
Use AN = 14 to get maximum value of AT: AT < 22; therefore maximum value = 21.
Positive difference = 21-3 = 18.
Now figure has finally appeared!
The rule is that the sum of the lengths of any two sides of a triangle is greater the length of the third. Thus considering triangle ANS first, we have:
5 + AN > 10 => AN > 5. Also AN < 15, hence 5 < AN < 15. Since only integral values are allowed, AN = {6, 7, 8,...,14}.
Using only the extremal values of AN in triangle ATN, we get:
Use AN = 6 to get minimum value of AT: AT + 6 > 8 => AT > 2; therefore minimum value = 3.
Use AN = 14 to get maximum value of AT: AT < 22; therefore maximum value = 21.
Positive difference = 21-3 = 18.
