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# If the lengths of all 3 sides of triangle RST are distinct, single-dig

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Joined: 02 Sep 2009
Posts: 58381
If the lengths of all 3 sides of triangle RST are distinct, single-dig  [#permalink]

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25 Feb 2019, 02:09
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Difficulty:

55% (hard)

Question Stats:

59% (01:50) correct 41% (01:43) wrong based on 69 sessions

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If the lengths of all 3 sides of triangle RST are distinct, single-digit prime numbers, then which of the following could be the perimeter of triangle RST ?

I. 14
II. 15
III. 19

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

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If the lengths of all 3 sides of triangle RST are distinct, single-dig  [#permalink]

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Updated on: 25 Feb 2019, 04:21
Bunuel wrote:
If the lengths of all 3 sides of triangle RST are distinct, single-digit prime numbers, then which of the following could be the perimeter of triangle RST ?

I. 14
II. 15
III. 19

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

prime single digit ; 2,3,5,7
so

15; 7+5+3 ; only possible since sum of two sides 5+3>7
for others
14 & 19 is not possible
IMO B

Originally posted by Archit3110 on 25 Feb 2019, 03:06.
Last edited by Archit3110 on 25 Feb 2019, 04:21, edited 1 time in total.
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Joined: 19 Jan 2019
Posts: 88
If the lengths of all 3 sides of triangle RST are distinct, single-dig  [#permalink]

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25 Feb 2019, 03:26
Archit3110 wrote:
Bunuel wrote:
If the lengths of all 3 sides of triangle RST are distinct, single-digit prime numbers, then which of the following could be the perimeter of triangle RST ?

I. 14
II. 15
III. 19

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

prime single digit ; 2,3,5,7
so
14; 7+5+2 ; yes
15; 7+5+3 ; yes
19; 7+7+5 ; yes , but no have to be distinct ; so no

Bunuel ; are given options correct?

The sides can be (2,3,5),(2,3,7), (3,5,7) and (2,5,7)...
And properties of triangle states that sum of two sides must always be greater than the third side.. keeping this in mind

Triangles with sides (2,3,5),(2,5,7) and (2,3,7) cannot be possible.. because 2+3 is equal to the third side.. and 2+5 is equal to 7 in the second triangle.. even 2+3<7 in the third triangle...

Leaving us with option (3,5,7) that satisfies 3+5>7, 5+7>3 and 3+7>5... Therefore 3+5+7 = 15 is the only possible solution..

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Re: If the lengths of all 3 sides of triangle RST are distinct, single-dig  [#permalink]

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03 Mar 2019, 19:52
1
Bunuel wrote:
If the lengths of all 3 sides of triangle RST are distinct, single-digit prime numbers, then which of the following could be the perimeter of triangle RST ?

I. 14
II. 15
III. 19

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

The single-digit prime numbers are 2, 3, 5 and 7. Thus, we could have 4 sets of 3 distinct, single-digit prime numbers:

{2, 3, 5}, {2, 3, 7}, {2, 5, 7} and {3, 5, 7}

However, in order for these sides to make a triangle, the triangle inequality tells us that the sum of the lengths of the two shortest sides must be greater than the length of the longest side. That makes {3, 5, 7} the only set whose members can be the side lengths of a triangle. Therefore, the perimeter can only be 3 + 5 + 7 = 15.

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Re: If the lengths of all 3 sides of triangle RST are distinct, single-dig  [#permalink]

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08 Mar 2019, 05:11
Pay attention to triangle properties. Sum of two sides is greater than third side.
Re: If the lengths of all 3 sides of triangle RST are distinct, single-dig   [#permalink] 08 Mar 2019, 05:11
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