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# If a triangle has sides 8, 10, and z, which of the following cannot be

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Math Expert
Joined: 02 Sep 2009
Posts: 49993
If a triangle has sides 8, 10, and z, which of the following cannot be  [#permalink]

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17 Sep 2018, 21:54
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Difficulty:

55% (hard)

Question Stats:

47% (01:20) correct 53% (01:02) wrong based on 59 sessions

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If a triangle has sides 8, 10, and z, which of the following cannot be the area of the triangle?

A. 3
B. 6
C. 12
D. 24
E. 46

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If a triangle has sides 8, 10, and z, which of the following cannot be  [#permalink]

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18 Sep 2018, 01:10
Bunuel wrote:
If a triangle has sides 8, 10, and z, which of the following cannot be the area of the triangle?

A. 3
B. 6
C. 12
D. 24
E. 46

Property : For given two sides of triangle and third side missing, the area of the triangle is maximum when the given two sides are perpendicular to each other. (Derivation attached)

i.e. $$A_{max.} = (1/2)*8*10 = 40$$

i.e. any number greater than 40 can NOT be area of this triangle

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Re: If a triangle has sides 8, 10, and z, which of the following cannot be  [#permalink]

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23 Sep 2018, 16:38
Bunuel wrote:
If a triangle has sides 8, 10, and z, which of the following cannot be the area of the triangle?

A. 3
B. 6
C. 12
D. 24
E. 46

If the triangle is a right triangle with z being the hypotenuse, then the two legs 10 and 8 will be the base and height of the triangle. In that case, the area of the triangle is ½ x 10 x 8 = 40. This is the maximum area for a triangle with two sides of 8 and 10; thus, any area less than 40 is a possible value for the area. Thus, answer choice E cannot be the area of the triangle.

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Re: If a triangle has sides 8, 10, and z, which of the following cannot be  [#permalink]

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27 Sep 2018, 01:00
When we know two of the side lengths, the greatest possible area of the triangle will occur when the two sides form a right triangle.

With $$8$$ and $$10$$ as the base and height of the triangle, we have an area of:

Area = 1⁄2 × Base × Height
$$Area = 1⁄2 × 8 × 10$$
$$Area = 40$$

Since the greatest possible area of the triangle with sides $$8$$, $$10$$, and $$z$$ is $$40$$,

other smaller areas ARE possible (if we change the angle to something other than 90 degrees).

Thus, it cannot have an area greater than $$40$$.

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Re: If a triangle has sides 8, 10, and z, which of the following cannot be  [#permalink]

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27 Sep 2018, 01:25
Bunuel wrote:
If a triangle has sides 8, 10, and z, which of the following cannot be the area of the triangle?

A. 3
B. 6
C. 12
D. 24
E. 46

Just use logic.
Think about it: given two sides and an unknown third - we can have a triangle with very very small area (almost 0), then the area increases and then finally starts decreasing again until it comes to 0 again. So there is a maximum limit to the area that the triangle can have. Hence given a range of increasing areas, the one which will not be possible is the largest one. For example, if 24 were not possible, 46 would not be possible too. But there is only one correct answer. Hence it must be 46 only.

Attachment:

TrianglesForm.jpeg [ 40.37 KiB | Viewed 206 times ]

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Re: If a triangle has sides 8, 10, and z, which of the following cannot be &nbs [#permalink] 27 Sep 2018, 01:25
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