DSQ: Calculating whether right triangles are similar - GMAT Quantitative

Assume Statement 1 alone. By the 45-45-90 Theorem, an isosceles right triangle has two acute angles that each measure ; specifically, and . By the Angle-Angle Similarity Postulate, .

Assume Statement 2 alone. The equality of the perimeters of two right triangles does not imply their similarity or nonsimilarity. For example:

Let

and

Since and , it follows by the Hypotenuse-Leg Theorem that , and, since congruent triangles are similar, it further follows that .

Now, let

and .

The sides are not in proportion, as , so .

In both scenarios, since , by the converse of the Pythagorean Theorem, the triangles are right, and the triangles both have perimeter , thereby satisfying the main premise and Statement 2. Statement 2 alone is therefore insufficient.

All right angles are congruent, so .

Since Statement 1 tells us that , this sets up the conditions for the Angle-Angle Similarity Postulate, so .

Statement 2 alone only tells us their hypotenuses. Congruence between one pair of angles and the measures of one pair of sides is insufficient information to determine whether two triangles are similar (given one angle, at least two pairs of proportional sides are required).

Therefore, the answer is that Statement 1 alone, but not Statement 2, is sufficient.

Assume both statements are true.

While in two similar triangles, the ratio of the perimeters, given in Statement 1, is indeed equal to that of the ratios of the lengths of the hypotenuses, given in Statement 2, this is not a sufficient condition for similarity. For example:

Case 1:

Case 2:

In each case, the conditions of the main problem and both statements are met, since:

Both triangles are right - each Pythagorean triple is a multiple of Pythagorean triple 3-4-5;

The ratio of the perimeters is ; and,

.

But in Case 1,

, since , and the similarity follows by way of the Side-Side-Side Similarity Principle.

In Case 2,

, since . This violates the conditions of similarity (note that in both cases, , but this is a different statement).

The two statements together are inconclusive.

Assume Statement 1 alone is true. Then, since , both being right angles, and from Statement 1, follows by way of the Angle-Angle Similarity Postulate. A similar argument shows Statement 2 also provides sufficient information.

The acute angles of a right triangle are complementary, so and are a complementary pair, as are and .

If Statement 1 is assumed—that is, if and are a complementary pair—then, since two angles complementary to the same angle—here, —must be congruent, . Since right angles , follows by way of the Angle-Angle Similarity Postulate, and Statement 1 turns out to provide sufficient information. By a similar argument, Statement 2 is also sufficient.

Assume Statement 1 alone. The statement used to find a sidelength ratio:

However, since we only know one sidelength ratio, similarity cannot be proved or disproved.

From Statement 2, another ratio can be found:

Again, since only one sidelength ratio is known, similarty can be neither proved nor disproved.

Assume both statements to be true. Similarity, by definition, requires that

From the two statements together, it can be seen that , so .

Assume Statement 1 alone. The ratio of the perimeters does not in and of itself establish similarity, since only one angle congruence is known.

Assume Statement 2 alone. The equation can be rewritten as a proportion statement:

This establishes that two pairs of corresponding sides are in proportion. Their included angles are both right angles, so , and follows from the Side-Angle-Side Similarity Theorem.

Each statement alone only gives a relationship between two sides within one triangle, so neither alone answers the question of the similarity of the two triangles.

Assume both statements are true. Then, since , .

By the multiplication property of inequality, since

and ,

Since, by definition, requires that , .

Assume Statement 1 alone. The area of the circle that circumscribes is four times the that of the circle that circumscribes , so the radius of the former is the square root of this, or two, times the radius of the latter; therefore, the ratio of the diameters of the circles is also two. Since in a right triangle inscribed inside a circle, the hypotenuse must be a diameter of that circle,

or

However, no other information is given or can be determined about any other sides or angles, so the triangles cannot been proved or disproved to be similar.

Assume Statement 2 alone. , or , but again, no other information is given or can be determined about any other sides or angles, so the triangles have not been proved or disproved to be similar.

Now assume both statements. , setting up the conditions for the Hypotenuse-Leg Similarity Theorem; it follows that .

Assume Statement 1 alone. and have the same degree measure, so the inscribed angles that intercept these arcs must also have the same degree measure - that is, . Since , both being right angles, this sets up the conditions of the Angle-Angle Postulate, so it follows that .

Assume Statement 2 alone. Major arc and major arc . by Statement 2, so

Again, the inscribed angles that intercept these arcs must also be congruent - that is, . Again, this, along with , prove that by way of the Angle-Angle Postulate.

Similarity of two triangles does not require that corresponding sides be congruent - only that they be in proportion. Therefore, the inequality of one pair of corresponding sides, as given in Statement 1 alone, is insufficient to prove or disprove that .

However, similarity of two triangles requires that all three pairs of corresponding angles be congruent. Therefore, Statement 2 alone, which establishes that one such pair is noncongruent, proves that .

Each of Statement 1 and Statement 2 gives information about only one of the triangles, so neither statement alone is sufficient.

Assume both statements are true. From Statement 1, and is right and measures .

From Statement 2 alone, is isosceles; the acute angles of an isosceles right triangle must both measure , so, in particular, . Also, it is given that is right.

and (both of the latter being right angles), and by the Angle-Angle Postulate, .

Assume both statements are true. Consider these two scenarios:

Let

and

Since and , then by the Hypotenuse-Leg Theorem, , and it follows that .

Now, let

and .

The sides are not in proportion, as , so .

In both scenarios, since , the triangles are right by the converse of the Pythagorean Theorem, satisfying the main premise; the triangles are scalene, having sides of three different lengths, satisfying Statement 1; and the triangles both have perimeter , satisfying Statement 2. The two statements together are insufficient.

Assume Statement 1 alone. If , then corresponding angles are congruent - specifically, . Statement 1 directly contradicts this, so .

Statement 2 alone does not prove or disprove similarity, since information is only given about one set of corresponding sides. No information is given about corresponding angles, or any other set of corresponding sides.

Assume Statement 1 alone. Since Quadrilateral is a trapezoid, it follows that ; by the Corresponding Angles Principle, . By reflexivity, . By the Angle-Angle Postulate, .

Assume Statement 2 alone.

Also, by reflexivity, . By the Side-Angle-Side Similarity Theorem, it follows that .

Assume Statement 1 alone. Rectangle is not a rhombus, so or . Rectangle is a rhombus, so . Therefore, either

or

Either way, the corresponding sides are not in proportion; it follows that .

Assume Statement 2 alone. No information is given about , so the quantities and cannot be proved to be equal or unequal. If they are unequal, since the sides are not in proportion, it follows that . If they are equal, then, since , it follows by the Side-Angle-Side similarity Theorem that .

Both rectangles, by definition, have four right angles.

A rhombus is a quadrilateral with four sides of equal length. A square is a quadrilateral with four right angles and four sides of equal length. Therefore, the two statements are actually equivalent in this context, so either they together provide insufficient information, or either alone does. We show that the latter is the case.

If both statements are true, and the rectangles are rhombuses and squares, then

and .

Consequently,

.

,

so by the Side-Angle-Side Similarity Theorem, .

Refer to the figure below, which gives the two rectangles with their diagonals as described.

Assume Statement 1 alone. The diagonals of a rectangle bisect the rectangle into congruent triangles, so and . Congruent triangles are also similar, so it follows that and . Since, by Statement 1, - or, stated differently, - by transitivity of similarity,

, and

.

Assume Statement 2 alone. The quadrilaterals are rectangles, so , both being right angles. From Statement 2, , setting up the conditions of the Angle-Angle Postulate; therefore, .

Assume both statements are true. We show the two statements together are insufficent to prove or disprove that

Suppose

.

Then Rectangle has perimeter

and area

Now set up two cases with different dimensions for Rectangle .

Case 1:

The perimeter of Rectangle is

,

one-third that of Rectangle .

The area of Rectangle is

,

one-ninth that of Rectangle .

,

so by the Side-Angle-Side Similarity Theorem, .

Case 2:

The perimeter of Rectangle is

,

one-third that of Rectangle .

The area of Rectangle is

,

one-ninth that of Rectangle .

Sides are not in proportion, making the statement false.

Assume Statement 1 alone. By the 45-45-90 Theorem, an isosceles right triangle has two acute angles that each measure ; specifically, and . By the Angle-Angle Similarity Postulate, .

Assume Statement 2 alone. The equality of the perimeters of two right triangles does not imply their similarity or nonsimilarity. For example:

Let

and

Since and , it follows by the Hypotenuse-Leg Theorem that , and, since congruent triangles are similar, it further follows that .

Now, let

and .

The sides are not in proportion, as , so .

In both scenarios, since , by the converse of the Pythagorean Theorem, the triangles are right, and the triangles both have perimeter , thereby satisfying the main premise and Statement 2. Statement 2 alone is therefore insufficient.