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Is the perimeter of a certain triangle less than 24 inches?

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Is the perimeter of a certain triangle less than 24 inches?  [#permalink]

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New post 22 Sep 2018, 08:07
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Is the perimeter of a certain triangle less than 24 inches?
(1) The shortest side of the triangle is 8\(\frac{1}{4}\) inches.
(2) The triangle is isosceles and the longest side is 1 inch longer than each of the other sides.

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Re: Is the perimeter of a certain triangle less than 24 inches?  [#permalink]

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New post 22 Sep 2018, 15:16
GMATbuster92 wrote:
Is the perimeter of a certain triangle less than 24 inches?
(1) The shortest side of the triangle is 8\(\frac{1}{4}\) inches.
(2) The triangle is isosceles and the longest side is 1 inch longer than each of the other sides.

\(0 < a \le b \le c\,\,\,\,\left[ {{\rm{inches}}} \right]\,\,\,\,\,\,\,\,\,\,\left( * \right)\)

\(a + b + c\,\,\mathop < \limits^? \,\,24\)

\(\left( 1 \right)\,\,\,a = 8{1 \over 4}\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,a + b + c\,\,\, \ge \,\,\,3 \cdot \left( {8{1 \over 4}} \right)\,\,\,\, > \,\,\,24\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle\)

\(\left( 2 \right)\,\,\,\left\{ \matrix{
a = b = x \hfill \cr
c = x + 1 \hfill \cr
\,\left( {\Delta \,\,{\rm{existence}}} \right)\,\,\,x + 1 < 2x\,\,\,\, \Leftrightarrow \,\,\,\,x > 1\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\,\left( {a,b,c} \right) = \left( {2,2,3} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\,\left( {a,b,c} \right) = \left( {8,8,9} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\,\,\,\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
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Re: Is the perimeter of a certain triangle less than 24 inches?  [#permalink]

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New post 23 Sep 2018, 02:12

Solution



To find:
    • Whether the perimeter of a certain triangle less than 24 inches or not

Analysing Statement 1
As per the information given in statement 1, the shortest side of the triangle is 8.25 inches
    • So, the other two sides are greater than 8.25 inches.
    • Hence, the perimeter can be 8.25 + (>8.25) + (>8.25) inches = greater than 24.75 inches

As we can conclude that perimeter is greater than 24 inches, statement 1 is sufficient.

Analysing Statement 2
As per the information given in statement 2, the triangle is isosceles, and the longest side is 1 inch longer than each of the other sides.
    • From this statement, we can’t find the exact length of the sides, and hence, can’t find the perimeter.

Therefore, statement 2 is not sufficient.

Hence, the correct answer is option A.

Answer: A
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Re: Is the perimeter of a certain triangle less than 24 inches? &nbs [#permalink] 23 Sep 2018, 02:12
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