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# In a triangle, three lines are drawn so that each line connects the

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Math Expert
Joined: 02 Sep 2009
Posts: 54370
In a triangle, three lines are drawn so that each line connects the  [#permalink]

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15 Aug 2017, 02:21
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Difficulty:

45% (medium)

Question Stats:

50% (01:31) correct 50% (01:27) wrong based on 44 sessions

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In a triangle, three lines are drawn so that each line connects the midpoint on one side with any point on an adjacent side. What is the greatest number of non-overlapping regions in the triangle that can be formed by the sides and the three lines?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

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In a triangle, three lines are drawn so that each line connects the  [#permalink]

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15 Aug 2017, 17:31
1
Bunuel wrote:
In a triangle, three lines are drawn so that each line connects the midpoint on one side with any point on an adjacent side. What is the greatest number of non-overlapping regions in the triangle that can be formed by the sides and the three lines?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Attachment:

triangleregions.jpg [ 22.45 KiB | Viewed 1228 times ]

Okay, I will take a stab at this one.

I think the answer is D.

I like patterns, a lot, which is great for, say, the logic games section of the LSAT (yup, did time as an attorney), but perhaps not so great in a math area where there is probably some Delphic decree about which I have never heard.

So I might be seeing a pattern where there is none.

In the diagram, one line makes two regions no matter how you draw it.

Two lines add two regions for a total of four regions.

As far as I can tell, three lines make 7 regions.

So the pattern seems to be:
1 -->2
2 -->4
3 -->7
where each line Lth adds L regions to whatever regions already existed. (Clumsily stated. Sorry.)

No lines, one region. Use line L = 1, and add one region to the one already there for a total of two.

One line, two regions -- add line L=2, add two regions to extant 2 to get total of four.

Two lines, four regions -- add Line L=3, add three regions to extant 4 to get total of seven.

I know there are at least 7. I drew them. If the pattern is not a figment of my imagination, then there is no way to make 8 regions with three lines.

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Re: In a triangle, three lines are drawn so that each line connects the  [#permalink]

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24 Aug 2017, 00:03
Bunuel wrote:
In a triangle, three lines are drawn so that each line connects the midpoint on one side with any point on an adjacent side. What is the greatest number of non-overlapping regions in the triangle that can be formed by the sides and the three lines?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Attachment:

IMG_20170824_123300165.jpg [ 24.96 KiB | Viewed 720 times ]

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Re: In a triangle, three lines are drawn so that each line connects the   [#permalink] 24 Aug 2017, 00:03
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