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

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Kudos [?]: 1274 [1] , given: 6002

Two sides of a triangle have lengths 6 and 8. Which of the f [#permalink]  30 Nov 2018, 17:28
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52% (00:44) correct 47% (01:30) wrong based on 19 sessions
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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Joined: 10 Apr 2015
Posts: 1795
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Kudos [?]: 1681 [1] , given: 8

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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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

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.

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

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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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