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Two sides of a triangle have lengths 6 and 8. Which of the f

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Two sides of a triangle have lengths 6 and 8. Which of the f [#permalink] New post 30 Nov 2018, 17:28
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Two sides of a triangle have lengths 6 and 8. Which of the following are possible areas of the triangle?

I. 2
II. 12
III. 24

A. I only

B. I and II only

C. II and III only

D. I and III only

E. I, II, and III
[Reveal] Spoiler: OA

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Re: Two sides of a triangle have lengths 6 and 8. Which of the f [#permalink] New post 01 Dec 2018, 15:12
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Carcass wrote:
Two sides of a triangle have lengths 6 and 8. Which of the following are possible areas of the triangle?

I. 2
II. 12
III. 24

A. I only

B. I and II only

C. II and III only

D. I and III only

E. I, II, and III


Area of triangle = (base)(height)/2

Let's let the side with length 8 be the base.

So, to MAXIMIZE the area of the triangle, we must MAXIMIZE the height of the triangle.
This occurs when the side with length 6 is PERPENDICULAR to the side with length 8
Image
In this case, area of triangle = (8)(6)/2 = 24
So, the greatest possible area = 24

Conversely, to MINIMIZE the area of the triangle, we must MINIMIZE the height of the triangle.
Well, if we keep decreasing the angle between the two given sides, we can make the height as small as we want.
Image
In fact, we could make the height equal 1 or 0.1 or even 0.0000000000000001

So, the area of the triangle must be greater than 0 and less than or equal to 24
In other words 0 < (area of triangle) ≤ 24

So, the area COULD be 2 or 12 or 24

Answer: E

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Re: Two sides of a triangle have lengths 6 and 8. Which of the f   [#permalink] 01 Dec 2018, 15:12
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Two sides of a triangle have lengths 6 and 8. Which of the f

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