DSQ: Calculating whether rectangles are similar - GMAT Quantitative
Card 1 of 48
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The statements are actually equivalent. Since 
, Statement 2 can be rewritten as 
, or 
- and, since all quantities are positive, 
. The question is therefore whether either statement alone answers the question or both together do not.
Each statement is equivalent to
.
Divide both sides by 
to yield a proportion statement:
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
The statements are actually equivalent. Since
Each statement is equivalent to
Divide both sides by
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
← Didn't Know|Knew It →
You are given two rectangles, 
and 
True or false: 
.
Statement 1: 
Statement 2: 
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, 
, 
, and, as a consequence, 
regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent,
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
← Didn't Know|Knew It →
You are given two rectangles, 
and 
True or false: 
.
Statement 1: 
Statement 2: 
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. Divide both sides by 
,
.
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
, 
,
and, regardless of whether the rectangles are similar or not,
.
Therefore, Statement 2 provides superfluous and unhelpful information.
Assume Statement 1 alone. Divide both sides by
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
and, regardless of whether the rectangles are similar or not,
Therefore, Statement 2 provides superfluous and unhelpful information.
← Didn't Know|Knew It →
You are given two rectangles, 
and 
.
Let the perimeter of 
be 
, and let the perimeter of 
be 
.
True or false: 
.
Statement 1: 
Statement 2: 
You are given two rectangles,
Let the perimeter of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The perimeter of 
is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
Assume Statement 1 alone. Then
, or 
, and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, 
.
By a similar argument, Statement 2 also proves .
The perimeter of
Similarly,
Therefore,
and, reducing,
Assume Statement 1 alone. Then
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
By a similar argument, Statement 2 also proves
← Didn't Know|Knew It →
You are given two rectangles, 
and 
.
Let the perimeter of 
be 
, and let the perimeter of 
be 
.
Let the area of 
be 
and the area 
be 
.
True or false: .
Statement 1: 
Statement 2: 
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is 
; the other two have the same dimensions, and both are called 
, except that the names of the vertices are differently arranged:
Regardless of which 
is chosen, the ratio of the perimeters is 
and the ratio of the areas is 
. The conditions of the problem are met for both pairings, but in one case, 
and in the other, 
(that is, the rectangles are similar, but the given similarity statement may or may not be true).
Assume Statement 2 alone.
The perimeter of 
is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
From Statement 2,
, or , and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, .
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is
Regardless of which
Assume Statement 2 alone.
The perimeter of
Similarly,
Therefore,
and, reducing,
From Statement 2,
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
← Didn't Know|Knew It →
You are given two rectangles, 
and 
.
Let the perimeter of 
be 
, and let the perimeter of 
be 
.
Let the area of 
be 
and the area 
be 
.
True or false: 
.
Statement 1: 
Statement 2: 
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true.
Examine these three rectangles. The one of the left is 
; the other two have the same dimensions, and both are called 
, except that the names of the vertices are differently arranged:
Regardless of which 
is chosen:
The ratio of the perimeters is 
;
The ratio of the areas is 
; and,
The ratio 
.
The conditions of the problem are met for both pairings, but in one case, 
and in the other, 
(that is, the rectangles are similar but the similarity statement given may be true or false).
The two statements together provide insufficient information.
Assume both statements to be true.
Examine these three rectangles. The one of the left is
Regardless of which
The ratio of the perimeters is
The ratio of the areas is
The ratio
The conditions of the problem are met for both pairings, but in one case,
The two statements together provide insufficient information.
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:.
Statement 1:
Statement 2:
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The statements are actually equivalent. Since , Statement 2 can be rewritten as , or - and, since all quantities are positive, . The question is therefore whether either statement alone answers the question or both together do not.
Each statement is equivalent to
.
Divide both sides by to yield a proportion statement:
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
The statements are actually equivalent. Since
Each statement is equivalent to
Divide both sides by
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
← Didn't Know|Knew It →
You are given two rectangles, and
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, , , and, as a consequence, regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent,
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
← Didn't Know|Knew It →
You are given two rectangles, and
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. Divide both sides by ,
.
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
, ,
and, regardless of whether the rectangles are similar or not,
.
Therefore, Statement 2 provides superfluous and unhelpful information.
Assume Statement 1 alone. Divide both sides by
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
and, regardless of whether the rectangles are similar or not,
Therefore, Statement 2 provides superfluous and unhelpful information.
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
Assume Statement 1 alone. Then
, or , and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, .
By a similar argument, Statement 2 also proves .
The perimeter of
Similarly,
Therefore,
and, reducing,
Assume Statement 1 alone. Then
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
By a similar argument, Statement 2 also proves
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
Let the area of be and the area be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:
Regardless of which is chosen, the ratio of the perimeters is and the ratio of the areas is . The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar, but the given similarity statement may or may not be true).
Assume Statement 2 alone.
The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
From Statement 2,
, or , and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, .
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is
Regardless of which
Assume Statement 2 alone.
The perimeter of
Similarly,
Therefore,
and, reducing,
From Statement 2,
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
Let the area of be and the area be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true.
Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:
Regardless of which is chosen:
The ratio of the perimeters is ;
The ratio of the areas is ; and,
The ratio .
The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar but the similarity statement given may be true or false).
The two statements together provide insufficient information.
Assume both statements to be true.
Examine these three rectangles. The one of the left is
Regardless of which
The ratio of the perimeters is
The ratio of the areas is
The ratio
The conditions of the problem are met for both pairings, but in one case,
The two statements together provide insufficient information.
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:.
Statement 1:
Statement 2:
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The statements are actually equivalent. Since , Statement 2 can be rewritten as , or - and, since all quantities are positive, . The question is therefore whether either statement alone answers the question or both together do not.
Each statement is equivalent to
.
Divide both sides by to yield a proportion statement:
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
The statements are actually equivalent. Since
Each statement is equivalent to
Divide both sides by
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
← Didn't Know|Knew It →
You are given two rectangles, and
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, , , and, as a consequence, regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent,
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
← Didn't Know|Knew It →
You are given two rectangles, and
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. Divide both sides by ,
.
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
, ,
and, regardless of whether the rectangles are similar or not,
.
Therefore, Statement 2 provides superfluous and unhelpful information.
Assume Statement 1 alone. Divide both sides by
This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Now examine Statement 2.
By the congruence of opposite sides of a rectangle,
and, regardless of whether the rectangles are similar or not,
Therefore, Statement 2 provides superfluous and unhelpful information.
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
Assume Statement 1 alone. Then
, or , and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, .
By a similar argument, Statement 2 also proves .
The perimeter of
Similarly,
Therefore,
and, reducing,
Assume Statement 1 alone. Then
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
By a similar argument, Statement 2 also proves
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
Let the area of be and the area be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:
Regardless of which is chosen, the ratio of the perimeters is and the ratio of the areas is . The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar, but the given similarity statement may or may not be true).
Assume Statement 2 alone.
The perimeter of is the sum of its sidelengths, and opposite sides are congruent, so
Similarly,
Therefore,
,
and, reducing,
From Statement 2,
, or , and by a property of proportions,
Therefore,
,
thereby proving the sides of the rectangles to be in proportion. As a consequence, .
Assume Statement 1 alone.
Examine these three rectangles. The one of the left is
Regardless of which
Assume Statement 2 alone.
The perimeter of
Similarly,
Therefore,
and, reducing,
From Statement 2,
Therefore,
thereby proving the sides of the rectangles to be in proportion. As a consequence,
← Didn't Know|Knew It →
You are given two rectangles, and .
Let the perimeter of be , and let the perimeter of be .
Let the area of be and the area be .
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
Let the perimeter of
Let the area of
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true.
Examine these three rectangles. The one of the left is ; the other two have the same dimensions, and both are called , except that the names of the vertices are differently arranged:
Regardless of which is chosen:
The ratio of the perimeters is ;
The ratio of the areas is ; and,
The ratio .
The conditions of the problem are met for both pairings, but in one case, and in the other, (that is, the rectangles are similar but the similarity statement given may be true or false).
The two statements together provide insufficient information.
Assume both statements to be true.
Examine these three rectangles. The one of the left is
Regardless of which
The ratio of the perimeters is
The ratio of the areas is
The ratio
The conditions of the problem are met for both pairings, but in one case,
The two statements together provide insufficient information.
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:.
Statement 1:
Statement 2:
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangle divided into two smaller rectangles.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
The statements are actually equivalent. Since , Statement 2 can be rewritten as , or - and, since all quantities are positive, . The question is therefore whether either statement alone answers the question or both together do not.
Each statement is equivalent to
.
Divide both sides by to yield a proportion statement:
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
The statements are actually equivalent. Since
Each statement is equivalent to
Divide both sides by
The sides of the rectangles are in proportion; subsequently, the rectangles are similar.
← Didn't Know|Knew It →
You are given two rectangles, and
True or false: .
Statement 1:
Statement 2:
You are given two rectangles,
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, , , and, as a consequence, regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent,
Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.
← Didn't Know|Knew It →