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# One side of a triangle has length 6 and a second side has length 7.

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Math Expert
Joined: 02 Sep 2009
Posts: 54368
One side of a triangle has length 6 and a second side has length 7.  [#permalink]

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09 Aug 2017, 04:45
00:00

Difficulty:

55% (hard)

Question Stats:

42% (02:17) correct 58% (01:43) wrong based on 47 sessions

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One side of a triangle has length 6 and a second side has length 7. Which of the following could be the area of this triangle?

I. 13
II. 21
III. 24

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

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Math Expert
Joined: 02 Aug 2009
Posts: 7556
Re: One side of a triangle has length 6 and a second side has length 7.  [#permalink]

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09 Aug 2017, 05:31
Bunuel wrote:
One side of a triangle has length 6 and a second side has length 7. Which of the following could be the area of this triangle?

I. 13
II. 21
III. 24

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

Hi...
Two points...
1. Area of triangle can be just slightly above 0 at the lower end.
2. The largest area will be right angle triangle with these sides

Ok lower end...Just above 0
Upper end... $$\frac{1}{2}*6*7=21$$
So 13 and 21 is possible, but 24 is NOT possible

D
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One side of a triangle has length 6 and a second side has length 7.  [#permalink]

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09 Aug 2017, 15:26
Nice one.
Here is what I did on this one -->

Side 1 -> 6
Side 2 -> 7

Maximum area would be when the two sides are perpendicular to each other forming a right triangle.

Hence Max. Area -> 1/2 * 6 * 7 => 21

Minimum area => When the third side would of length zero and the triangle would cease to exist.

Hence the range of Areas => [0,21]

Hence Option 1 and Option 2 are possible.
Option 3 is not.

Hence D.

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One side of a triangle has length 6 and a second side has length 7.  [#permalink]

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09 Aug 2017, 18:52
stonecold wrote:
Nice one.
Here is what I did on this one -->

Side 1 -> 6
Side 2 -> 7

Maximum area would be when the two sides are perpendicular to each other forming a right triangle.

Hence Max. Area -> 1/2 * 6 * 7 => 21

Minimumm area => When the third side would of length zero and the triangle would cease to exist.

Hence the range of Areas => [0,21]

Hence Option 1 and Option 2 are possible.
Option 3 is not.

Hence D.

stonecold, I was waiting for this comment so I could ask somebody, so thank you: are you talking about a degenerate triangle? The kind that looks like a line segment?

And if so, have you ever seen an official GMAT question that expects us to know that there is actually such a thing as a triangle with area of zero?
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One side of a triangle has length 6 and a second side has length 7.  [#permalink]

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09 Aug 2017, 21:04
Bunuel wrote:
One side of a triangle has length 6 and a second side has length 7. Which of the following could be the area of this triangle?

I. 13
II. 21
III. 24

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

The triangle will have a maximum area when it is a right angled triangle
The largest triangle will be formed when 6 and 7 are the base and height of triangle respectively or vice versa.
Max Area = 1/2 * 7 * 6 = 21
There is no lower limit on the area of any triangle whose only 2 sides are known. Hence 21 => Area > 0
So we can also have triangle with area = 13 but not with area 24
Hence D
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One side of a triangle has length 6 and a second side has length 7.   [#permalink] 09 Aug 2017, 21:04
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