SSAT Upper Level Quantitative › How to find if two acute / obtuse triangles are similar
;
;
has perimeter 400.
Which of the following is equal to ?
The perimeter of is actually irrelevant to this problem. Corresponding sides of similar triangles are in proportion, so use this to calculate
, or
:
;
;
;
has perimeter 90.
Give the perimeter of .
The ratio of the perimeters of two similar triangles is the same as the ratio of the lengths of a pair of corresponding sides. Therefore,
Given: and
;
and
.
Which of the following statements would not be enough, along with what is given, to prove that ?
The given information is enough to prove the triangles similar.
Two pairs of corresponding angles are stated to be congruent in the main body of the problem; it follows from the Angle-Angle Similarity Postulate that the triangles are similar. No further information is needed.
Given: and
;
.
Which of the following statements would not be enough, along with what is given, to prove that ?
The given information is enough to prove the triangles similar.
From both the given proportion statement and either or
, it follows that
—all three pairs of corresponding sides are in proportion; by the Side-Side-Side Similarity Theorem,
. From the given proportion statement and
, since these are the included angles of the sides that are in proportion, then by the Side-Angle-Side Similarity Theorem,
. From the given proportion statement and
, since these are nonincluded angles of the sides that are in proportion, no similarity can be deduced.
.
Evaluate .
These triangles cannot exist.
The similarity of the triangles is actually extraneous information here. The sum of the measures of a triangle is , so:
;
;
has perimeter 300.
Evaluate .
Insufficient information is given to answer the problem.
The ratio of the perimeters of two similar triangles is equal to the ratio of the lengths of a pair of corresponding sides. Therefore,
and
, or
By one of the properties of proportions, it follows that
The perimeter of is
, so
;
Which of the following is true about ?
is isosceles and acute.
is scalene and obtuse.
is isosceles and obtuse.
is scalene and acute.
None of the other responses is correct.
Corresponding angles of similar triangles are congruent, so the measures of the angles of are equal to those of
.
, so
. Also
, so
.
All three angles have measure less than , so
is acute. Also, two of the angles are congruent, so by the Converse of the Isosceles Triangle Theorem,
is isosceles.
;
.
Which of the following is true about ?
is isosceles and acute.
is isosceles and obtuse.
is scalene and acute.
is scalene and obtuse.
None of the other responses is correct.
, so corresponding sides are in proportion; it follows that
Therefore, is isosceles.
Also, corresponding angles are congruent, so if acute (or obtuse), so is
. We can compare the sum of the squares of the lesser two sides to that of the greatest;
The sum of the squares of the lesser sides is greater than the square of the greatest side, so is acute - and so is
. The correct response is that
is isosceles and acute.
;
;
.
Which of the following correctly gives the relationship of the angles of ?
Corresponding angles of similar triangles are congruent, so .
Consequently,
Therefore,
.
;
Which of the following is true about ?
is scalene and obtuse.
is scalene and acute.
is isosceles and obtuse.
is isosceles and acute.
None of the other responses is correct.
Corresponding angles of similar triangles are congruent, so the measures of the angles of are equal to those of
.
Two of the angles of have measures
and
; its third angle measures
.
One of the angles having measure greater than makes
- and, consequently,
- an obtuse triangle. Also, the three angles have different measures, so the sides do as well, making
scalene.