Calculating whether rectangles are similar - GMAT Quantitative
Card 1 of 48
In the above diagram,
.
and 
. Give the area of 
.
In the above diagram,
Tap to reveal answer
, so
The area of the rectangle is
The area of the rectangle is
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, and 
.
All the following quantities MUST be equal to 2 except for .
All the following quantities MUST be equal to 2 except for .
Tap to reveal answer
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
.
Since opposite sides of a rectangle are congruent, 
, so
.
and 
are diagonals of the rectangle. If they are constructed, then, since 
and 
(both are right angles), by the Side-Angle-Side Similarity Theorem, 
. By similarity, 
.
The ratio of the perimeters of the rectangles is
,
It follows from 
and a property of proportions that this ratio is equal to 
.
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
Since opposite sides of a rectangle are congruent,
The ratio of the perimeters of the rectangles is
It follows from
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
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;
.
Which of the following must be equal to 
?
Which of the following must be equal to
Tap to reveal answer
, so 
; this is the similarity ratio of 
to 
.
and 
are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that 
.
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not 
.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is 
.
However, by similarity, and the fact that opposite sides 
and 
are congruent,
.
The correct choice is 
.
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is
However, by similarity, and the fact that opposite sides
The correct choice is
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Note: Figure NOT drawn to scale.
Refer to the above figure. 
.
.
Give the ratio of the area of the shaded region to the area of 
.
Note: Figure NOT drawn to scale.
Refer to the above figure.
Give the ratio of the area of the shaded region to the area of
Tap to reveal answer
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of 
to that of 
is
.
So if the area of 
is 
, the area of 
is 
The area of the shaded region is the difference between the areas of the rectangles, making this area
.
The desired ratio is 24 to 25.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of
So if the area of
The area of the shaded region is the difference between the areas of the rectangles, making this area
The desired ratio is 24 to 25.
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A certain rectangle has a length of 
and a width of 
. Which of the following dimensions would another rectangle need in order for it to be similar?
A certain rectangle has a length of
Tap to reveal answer
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of 
and a width of 
has the same ratio of dimensions as a rectangle with a length of 
and a width of 
, so these two rectangles are similar.
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of
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An engineer is making a scale model of a building. The real building needs to have a width of 
and a length of 
. If the engineer's scale model has a width of 
, what does the length of the model need to be?
An engineer is making a scale model of a building. The real building needs to have a width of
Tap to reveal answer
An engineer is making a scale model of a building. The real building needs to have a width of 
and a length of 
. If the engineer's scale model has a width of 
, what does the length of the model need to be?
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
An engineer is making a scale model of a building. The real building needs to have a width of
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
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In the above diagram,
.
and . Give the area of .
In the above diagram,
Tap to reveal answer
, so
The area of the rectangle is
The area of the rectangle is
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, and .
All the following quantities MUST be equal to 2 except for .
All the following quantities MUST be equal to 2 except for .
Tap to reveal answer
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
.
Since opposite sides of a rectangle are congruent, , so
.
and are diagonals of the rectangle. If they are constructed, then, since and (both are right angles), by the Side-Angle-Side Similarity Theorem, . By similarity, .
The ratio of the perimeters of the rectangles is
,
It follows from and a property of proportions that this ratio is equal to .
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
Since opposite sides of a rectangle are congruent,
The ratio of the perimeters of the rectangles is
It follows from
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
← Didn't Know|Knew It →
;
.
Which of the following must be equal to ?
Which of the following must be equal to
Tap to reveal answer
, so ; this is the similarity ratio of to .
and are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that .
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not .
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is .
However, by similarity, and the fact that opposite sides and are congruent,
.
The correct choice is .
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is
However, by similarity, and the fact that opposite sides
The correct choice is
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Note: Figure NOT drawn to scale.
Refer to the above figure. .
.
Give the ratio of the area of the shaded region to the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure.
Give the ratio of the area of the shaded region to the area of
Tap to reveal answer
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of to that of is
.
So if the area of is , the area of is
The area of the shaded region is the difference between the areas of the rectangles, making this area
.
The desired ratio is 24 to 25.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of
So if the area of
The area of the shaded region is the difference between the areas of the rectangles, making this area
The desired ratio is 24 to 25.
← Didn't Know|Knew It →
A certain rectangle has a length of and a width of . Which of the following dimensions would another rectangle need in order for it to be similar?
A certain rectangle has a length of
Tap to reveal answer
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of and a width of has the same ratio of dimensions as a rectangle with a length of and a width of , so these two rectangles are similar.
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of
← Didn't Know|Knew It →
An engineer is making a scale model of a building. The real building needs to have a width of and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?
An engineer is making a scale model of a building. The real building needs to have a width of
Tap to reveal answer
An engineer is making a scale model of a building. The real building needs to have a width of and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
An engineer is making a scale model of a building. The real building needs to have a width of
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
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Note: Figure NOT drawn to scale.
Refer to the above figure. .
.
Give the ratio of the area of the shaded region to the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure.
Give the ratio of the area of the shaded region to the area of
Tap to reveal answer
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of to that of is
.
So if the area of is , the area of is
The area of the shaded region is the difference between the areas of the rectangles, making this area
.
The desired ratio is 24 to 25.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of
So if the area of
The area of the shaded region is the difference between the areas of the rectangles, making this area
The desired ratio is 24 to 25.
← Didn't Know|Knew It →
In the above diagram,
.
and . Give the area of .
In the above diagram,
Tap to reveal answer
, so
The area of the rectangle is
The area of the rectangle is
← Didn't Know|Knew It →
, and .
All the following quantities MUST be equal to 2 except for .
All the following quantities MUST be equal to 2 except for .
Tap to reveal answer
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
.
Since opposite sides of a rectangle are congruent, , so
.
and are diagonals of the rectangle. If they are constructed, then, since and (both are right angles), by the Side-Angle-Side Similarity Theorem, . By similarity, .
The ratio of the perimeters of the rectangles is
,
It follows from and a property of proportions that this ratio is equal to .
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
Since opposite sides of a rectangle are congruent,
The ratio of the perimeters of the rectangles is
It follows from
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
← Didn't Know|Knew It →
;
.
Which of the following must be equal to ?
Which of the following must be equal to
Tap to reveal answer
, so ; this is the similarity ratio of to .
and are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that .
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not .
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is .
However, by similarity, and the fact that opposite sides and are congruent,
.
The correct choice is .
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is
However, by similarity, and the fact that opposite sides
The correct choice is
← Didn't Know|Knew It →
A certain rectangle has a length of and a width of . Which of the following dimensions would another rectangle need in order for it to be similar?
A certain rectangle has a length of
Tap to reveal answer
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of and a width of has the same ratio of dimensions as a rectangle with a length of and a width of , so these two rectangles are similar.
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of
← Didn't Know|Knew It →
An engineer is making a scale model of a building. The real building needs to have a width of and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?
An engineer is making a scale model of a building. The real building needs to have a width of
Tap to reveal answer
An engineer is making a scale model of a building. The real building needs to have a width of and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
An engineer is making a scale model of a building. The real building needs to have a width of
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
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Note: Figure NOT drawn to scale.
Refer to the above figure. .
.
Give the ratio of the area of the shaded region to the area of .
Note: Figure NOT drawn to scale.
Refer to the above figure.
Give the ratio of the area of the shaded region to the area of
Tap to reveal answer
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of to that of is
.
So if the area of is , the area of is
The area of the shaded region is the difference between the areas of the rectangles, making this area
.
The desired ratio is 24 to 25.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of
So if the area of
The area of the shaded region is the difference between the areas of the rectangles, making this area
The desired ratio is 24 to 25.
← Didn't Know|Knew It →
In the above diagram,
.
and . Give the area of .
In the above diagram,
Tap to reveal answer
, so
The area of the rectangle is
The area of the rectangle is
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