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Is the perimeter of triangle ABC greater than 12 cm?

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by AAPL » Sat Dec 08, 2018 1:51 pm

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e-GMAT

Is the perimeter of triangle ABC greater than 12 cm?

1) One side of triangle ABC measures 6 cm.
2) The lengths of the three sides of triangle ABC are consecutive positive even integers.

OA D
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Is the perimeter of triangle ABC greater than 12 cm?

by fskilnik@GMATH » Sun Dec 09, 2018 7:08 am
AAPL wrote:e-GMAT

Is the perimeter of triangle ABC greater than 12 cm?

1) One side of triangle ABC measures 6 cm.
2) The lengths of the three sides of triangle ABC are consecutive positive even integers.
All lengths are measured in centimeters.
$$a + b + c\,\,\mathop > \limits^? \,\,12$$
$$\left( 1 \right)\,\,{\rm{WLOG}}\,\left( * \right)\,\,\,a = 6\,\,\,\,\,\mathop \Rightarrow \limits^{\Delta \,\,{\rm{exists!}}} \,\,\,\,6 = a < b + c\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,$$
(*) WLOG = without loss of generality

$$\eqalign{
& \left( 2 \right)\,\,a + b + c = 2M + 2M + 2 + 2M + 4 = 6M + 6\,\,,\,\,\,M \ge 2\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \cr
& \left( {M = 1\,\,\, \Rightarrow \,\,\,\,2,4,6\,\,{\rm{is}}\,\,{\rm{not}}\,\,{\rm{a}}\,\,{\rm{triangle!}}} \right) \cr} $$
(We have NOT said that a, b and c are respectively equal to 2M, 2M+2 and 2M+4 ... be careful!)


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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Jay@ManhattanReview » Sun Dec 09, 2018 8:51 pm
AAPL wrote:e-GMAT

Is the perimeter of triangle ABC greater than 12 cm?

1) One side of triangle ABC measures 6 cm.
2) The lengths of the three sides of triangle ABC are consecutive positive even integers.

OA D
Say the three sides are a, b and c.

Thus, perimeter = a + b + c.

We have to ascertain whether a + b + c > 12.

Let's take each statement one by one.

1) One side of triangle ABC measures 6 cm.

Note that the sum of any two sides is greater than the third side.

Say a = 6, thus, b + c > 6

=> a + b + c > 12. Sufficient.

2) The lengths of the three sides of triangle ABC are consecutive positive even integers.

Case 1: Say the three sides are 2, 4, and 6, minimum possible consecutive positive even integers. However, this combination of sides will not render a triangle since the sum of the smaller sides = 2 + 4 = 6 is not greater than the third side ( = 6).

Case 2: Say the three sides are 4, 6, and 8, next possible consecutive positive even integers. This combination of the sides renders a triangle since the sum of the smaller sides = 4 + 6 = 10 > third side (= 8). Perimeter = 4 + 6 + 8 = 18 > 12. Sufficient.

There is no need to check other cases such as 6, 8 and 10; 8, 10, and 12, etc. since they all form triangles and perimeter > 12. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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