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Two congruent triangles were combined in three different ways, as show

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Two congruent triangles were combined in three different ways, as show  [#permalink]

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New post 18 Sep 2016, 10:02
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Two congruent triangles were combined in three different ways, as shown above. If the perimeters of the shapes above are 12, 14, and 16, then what is the perimeter of one triangle?

A. 13/2
B. 14/2
C. 21/2
D. 22/2
E. 42/3

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Re: Two congruent triangles were combined in three different ways, as show  [#permalink]

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New post 18 Sep 2016, 18:47
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Let sides of each of the triangle be a, b and c.
Perimeters of the combined shapes are given. In each of the shapes, only two sides contribute to the perimeter as the third side is in the interior.

\(2(a+b) =12\)
\(2(b+c) = 14\)
\(2(c+a) = 16\)

Adding, \(4 (a+b+c) = 42\)
\(a+b+c = \frac{21}{2}\)

Option C
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Two congruent triangles were combined in three different ways, as show  [#permalink]

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New post 19 Sep 2016, 05:11
Given that the two triangles are congruent we can name their sides as a, b and c. It doesn't really matter how you choose to label the sides, but let's go with a=height (short base), b=long base, c=hypotenuse.

Based on this the perimeter of the shapes resulting from combining the triangles in different ways (as shown in the figure) is

Rectangle (two long bases, two heights): 2a + 2b = 12,

Parallelogram on top right of the figure (two short heights, two hypotenuses): 2a + 2c = 14 and

Bottom parallelogram (two long bases, two hypotenuses): 2b + 2c = 16.

We could use these three equations to find the individual values of a,b and c, but we don't really need to - we're looking for the perimeter of the triangle, which is the expression a+b+c. To find the value of that expression, add all three equations to get

4a + 4b + 4c = 42

--> 4(a + b + c) = 42

--> a + b + c = 42/4 = 21/2.

Hence, the perimeter of one triangle is 21/2.
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Re: Two congruent triangles were combined in three different ways, as show  [#permalink]

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New post 18 Mar 2019, 01:59
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Re: Two congruent triangles were combined in three different ways, as show   [#permalink] 18 Mar 2019, 01:59
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