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M06-09

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Intern
Joined: 06 Jul 2013
Posts: 43
Location: United States

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11 Aug 2018, 21:00
Bunuel wrote:
Official Solution:

What is the area of a triangle with the following vertices $$L(1, 3)$$, $$M(5, 1)$$, and $$N(3, 5)$$?

A. 3
B. 4
C. 5
D. 6
E. 7

Make a diagram:

Notice that the area of the blue square is $$4^2=16$$ and the area of the red triangle is 16 minus the areas of 3 little triangles which are in the corners ($$2*\frac{2}{2}$$, $$4*\frac{2}{2}$$ and $$4*\frac{2}{2}$$). Therefore, the area of a triangle LMN is $$16-(2+4+4)=6$$.

Hi Bunuel,

Why can't the height be 4 and the base be 4 as well? Height being the base of the square to point N, and the base being point M to the left side of the square? So b*h/2 or 4*4/2 = 8
Math Expert
Joined: 02 Sep 2009
Posts: 49430

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12 Aug 2018, 01:57
TippingPoint93 wrote:
Bunuel wrote:
Official Solution:

What is the area of a triangle with the following vertices $$L(1, 3)$$, $$M(5, 1)$$, and $$N(3, 5)$$?

A. 3
B. 4
C. 5
D. 6
E. 7

Make a diagram:

Notice that the area of the blue square is $$4^2=16$$ and the area of the red triangle is 16 minus the areas of 3 little triangles which are in the corners ($$2*\frac{2}{2}$$, $$4*\frac{2}{2}$$ and $$4*\frac{2}{2}$$). Therefore, the area of a triangle LMN is $$16-(2+4+4)=6$$.

Hi Bunuel,

Why can't the height be 4 and the base be 4 as well? Height being the base of the square to point N, and the base being point M to the left side of the square? So b*h/2 or 4*4/2 = 8

Any side of a triangle can be considered as base and similarly any perpendicular from the opposite vertex to the corresponding base can be considered as height. So, for example, if you consider ML to be base, then the height would be perpendicular from N to ML. None of the sides equals 4 in the triangle MNL and none of the heights is equal to 4. That's why your way is not correct.
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Director
Joined: 30 Jan 2016
Posts: 716
Location: United States (MA)

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Updated on: 08 Sep 2018, 01:36
Another approach is to use Pick's formula:

Area= M/2 + N - 1,

where
M of lattice points on the boundary placed on the polygon's perimeter
N - lattice points in the interior located in the polygon

M - green dots;
N - blue dots

Area = 6/3 +4 - 1 = 3+4-1 = 6
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Non progredi est regredi

Originally posted by Akela on 08 Sep 2018, 01:28.
Last edited by Akela on 08 Sep 2018, 01:36, edited 1 time in total.
Director
Joined: 30 Jan 2016
Posts: 716
Location: United States (MA)

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08 Sep 2018, 01:33
Find the area of the following polygon, if the grid is 1 cm X 1cm
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Non progredi est regredi

Re: M06-09 &nbs [#permalink] 08 Sep 2018, 01:33

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