Calculating if two acute / obtuse triangles are similar - GMAT Quantitative Reasoning
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The triangles are similar. What is the value of x?
The triangles are similar. What is the value of x?
The proportions of corresponding sides of similar triangles must be equal. Therefore, 
. 
.
The proportions of corresponding sides of similar triangles must be equal. Therefore,
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is an equilateral triangle. Points 
are the midpoints of 
, respectively. 
is constructed.
All of the following are true except:
All of the following are true except:
The three sides of 
are the midsegments of 
, so 
is similar to 
.
By the Triangle Midsegment Theorem, each is parallel to one side of 
.
By the same theorem, each has length exactly half of that side, giving 
twice the perimeter of 
.
But since the sides of 
have twice the length of those of 
, the area of 
, which varies directly as the square of a sidelength, must be four times that of 
.
The correct choice is the one that asserts that the area of 
is twice that of 
.
The three sides of
By the Triangle Midsegment Theorem, each is parallel to one side of
By the same theorem, each has length exactly half of that side, giving
But since the sides of
The correct choice is the one that asserts that the area of
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.
Order the angles of 
from least to greatest measure.
Order the angles of
In a triangle, the angle of greatest measure is opposite the side of greatest measure, and the angle of least measure is opposite the side of least measure. 
, so their opposite angles are ranked in this order - that is, 
.
Corresponding angles of similar triangles are congruent, so, since 
, 
.
Therefore, by substitution, 
.
In a triangle, the angle of greatest measure is opposite the side of greatest measure, and the angle of least measure is opposite the side of least measure.
Corresponding angles of similar triangles are congruent, so, since
Therefore, by substitution,
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