DSQ: Calculating the perimeter of an acute / obtuse triangle - GMAT Quantitative
Card 1 of 104
True or false: The perimeter of 
is greater than 50.
Statement 1: 
is an isosceles triangle.
Statement 2: 
and 
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
We can find the minimum of the perimeter:
:
The perimeter of 
is greater than 50.
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
We can find the minimum of the perimeter:
The perimeter of
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Find the perimeter of the obtuse 
.
I) 
.
II) 
.
Find the perimeter of the obtuse
I)
II)
Tap to reveal answer
We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.
I) Relates the two shorter sides.
II) Relates the longest side to one of the short sides.
However, we cannot find any of our side lengths, so we cannot find the perimeter.
We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.
I) Relates the two shorter sides.
II) Relates the longest side to one of the short sides.
However, we cannot find any of our side lengths, so we cannot find the perimeter.
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Give the perimeter of 
.
Statement 1: 
Statement 2: 
Give the perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
We demonstrate that both statements together provide insufficient information by examining two cases:
Case 1: 
The perimeter of 
is
.
Case 2: 
.
The perimeter of 
is
.
Both cases satisfy the conditions of both statements but different perimeters are yielded.
We demonstrate that both statements together provide insufficient information by examining two cases:
Case 1:
The perimeter of
Case 2:
The perimeter of
Both cases satisfy the conditions of both statements but different perimeters are yielded.
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True or false: The perimeter of 
is greater than 24.
Statement 1: 
Statement 2: 
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:
Case 1: 
This triangle satisfies the triangle inequality, since 
; its perimeter is 
Case 2: 
This triangle satisfies the triangle inequality, since 
; its perimeter is 
.
Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.
Assume Statement 2 alone. Again, 
; since, by Statement 2, 
, by substitution, 
. The perimeter of 
is
, and, since 
, then
The perimeter of 
is greater than 24.
Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:
Case 1:
This triangle satisfies the triangle inequality, since
Case 2:
This triangle satisfies the triangle inequality, since
Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.
Assume Statement 2 alone. Again,
The perimeter of
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True or false: The perimeter of 
is greater than 50.
Statement 1: 
is an isosceles triangle.
Statement 2: 
and 
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
Also,
Therefore,
,
and we can find the range of the values of the perimeter
by adding:
Therefore, the perimeter may or may not be greater than 50.
Assume both statements to be true. An isosceles triangle has two sides of the same length, so either 
or 
.
If 
, the perimeter is
If 
, the perimeter is
Either way, the perimeter is less than 50.
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
Also,
Therefore,
and we can find the range of the values of the perimeter
by adding:
Therefore, the perimeter may or may not be greater than 50.
Assume both statements to be true. An isosceles triangle has two sides of the same length, so either
If
If
Either way, the perimeter is less than 50.
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Give the perimeter of 
.
Statement 1: 
Statement 2: 
Give the perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
The perimeter of 
is equal to the sum of the lengths of the sides; that is, 
.
From Statement 1 alone, we get
we can add 
to both sides to get
However, without any further information, we cannot determine the actual perimeter.
A similar argument shows that Statement 2 alone gives insufficient information as well.
However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:
Divide both sides by 3:
Since
,
we can substitute 29 for 
and find 
:
While we cannot find 
or 
individually, this is not necessary; in the perimeter formula, we can substitute 29 for 
and 8 for 
:
.
The perimeter of
From Statement 1 alone, we get
we can add
However, without any further information, we cannot determine the actual perimeter.
A similar argument shows that Statement 2 alone gives insufficient information as well.
However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:
Divide both sides by 3:
Since
we can substitute 29 for
While we cannot find
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True or false: The perimeter of 
is greater than 60.
Statement 1: 
is an isosceles triangle.
Statement 2: 
and 
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true. An isosceles triangle has two sides of equal length, so either 
or 
. By the Triangle Inequality Theorem, the length of each side must be less than the sum of the lengths of the other two; both scenarios are possible, since
and
.
If 
, the perimeter of 
is
.
If 
, the perimeter of 
is
.
Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.
Assume both statements to be true. An isosceles triangle has two sides of equal length, so either
and
If
If
Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.
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What is the perimeter of 
?
Statement 1: The triangle with its vertices at the midpoints of 
, 
, and 
has perimeter 34.
Statement 2: 
, 
, and 
are the midpoints of the sides of a triangle with perimeter 136.
What is the perimeter of
Statement 1: The triangle with its vertices at the midpoints of
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. A segment that has as its endpoints the midpoints of two sides of a triangle is a midsegment of the triangle, and its length is half that of the side to which it is parallel. Therefore, the sum of the lengths of the midsegments—that is, the perimeter of the triangle they form, which from Statement 1 is 34, is half the perimeter of the larger triangle, which here is 
. The perimeter of 
is therefore 68.
Assume Statement 2 alone. Here, 
itself is the triangle formed by the midsegments. Since the larger triangle has perimeter 136, 
has perimeter half this, or 68.
Assume Statement 1 alone. A segment that has as its endpoints the midpoints of two sides of a triangle is a midsegment of the triangle, and its length is half that of the side to which it is parallel. Therefore, the sum of the lengths of the midsegments—that is, the perimeter of the triangle they form, which from Statement 1 is 34, is half the perimeter of the larger triangle, which here is
Assume Statement 2 alone. Here,
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True or false: 
and 
have the same perimeter.
Statement 1: is isosceles and is scalene.
Statement 2: 
and 
.
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 alone provides insufficient information to answer the question, since it is possible for an isosceles triangle, which has two or three sides of equal length, to have perimeter equal to or not equal to a scalene triangle, which has three sides of different lengths. For example, a triangle with sides of length 10, 10, and 12 has perimeter 
, the same as a triangle with sides of length 9, 10, and 13, since 
, but a triangle with sides of length 10, 10, and 13 has perimeter 
.
Statement 2 alone provides insufficient information to answer the question. Since 
and 
, it follows that the perimeter are equal if and only if 
; we are not told whether this is true or false.
Now assume both statements. is isosceles, so two of its sides have equal length; however, it cannot hold that 
; if so, then, since 
and 
, it would follow that 
, which contradicts being scalene. Therefore, either 
or 
. If 
, then 
is congruent to one other side of , and, consequently, one other side of , contradicting being scalene. Therefore, 
; as stated before, the perimeters are equal if and only if 
, so the perimeters are not equal.
Statement 1 alone provides insufficient information to answer the question, since it is possible for an isosceles triangle, which has two or three sides of equal length, to have perimeter equal to or not equal to a scalene triangle, which has three sides of different lengths. For example, a triangle with sides of length 10, 10, and 12 has perimeter
Statement 2 alone provides insufficient information to answer the question. Since
Now assume both statements.
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Given and Square 
, which one has the greater perimeter?
Statement 1: 
Statement 2: 
Given
Statement 1:
Statement 2:
Tap to reveal answer
For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the sidelength. The perimeter of the square is, as a result, 4, and the length of each of the diagonals 
and 
is 
times the length of a side, or simply 
.
The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as
Statement 1: 
- or equivalently, 
Statement 2: 
Assume Statement 1 alone. By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. Therefore,
and
Since from Statement 1, , t
By the Addition Property of Inequality, we can add 
to both sides;
The perimeter of the triangle is greater than 4; equivalently, has greater perimeter than Square 
.
Assume Statement 2. By similar reasoning, since one side has length 
, the perimeter is at greater than twice this, or 
, which is greater than 4, so has greater perimeter than Square 
.
For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the sidelength. The perimeter of the square is, as a result, 4, and the length of each of the diagonals
The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as
Statement 1:
Statement 2:
Assume Statement 1 alone. By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. Therefore,
and
Since from Statement 1,
By the Addition Property of Inequality, we can add
The perimeter of the triangle is greater than 4; equivalently,
Assume Statement 2. By similar reasoning, since one side has length
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Given Triangle and Square 
, which one has the greater perimeter?
Statement 1: 
Statement 2: 
Given Triangle
Statement 1:
Statement 2:
Tap to reveal answer
For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the side length. The perimeter of the square is, as a result, 4, and the length of each of the diagonals 
and 
is 
times the length of a side, or simply 
.
The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as
Statement 1: 
Statement 2: 
We show that these two statements together provide insufficient information.
By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. We can get the range of values of 
using this fact:
Also,
So,
Add 
and 
to all three expressions; the expression in the middle is the perimeter of :
Since 
, for all practical purposes,
Therefore, we cannot tell whether the perimeter is less than, equal to, or greater than 4. Equivalently, we cannot determine whether the triangle or the square has the greater perimeter.
For the sake of simplicity, we will assume the length of each side of the square is 1; this reasoning works independently of the side length. The perimeter of the square is, as a result, 4, and the length of each of the diagonals
The equivalent question becomes whether the perimeter of the triangle is greater than, equal to, or less than 4. The statements can be rewritten as
Statement 1:
Statement 2:
We show that these two statements together provide insufficient information.
By the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third. We can get the range of values of
Also,
So,
Add
Since
Therefore, we cannot tell whether the perimeter is less than, equal to, or greater than 4. Equivalently, we cannot determine whether the triangle or the square has the greater perimeter.
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Given: and 
, with 
and 
.
True or false: and 
have the same perimeter.
Statement 1: 
Statement 2: 
Given:
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
Assume Statement 1 alone. We show that this provides insufficient information by examining two scenarios.
Case 1: 
. By definition, and , satisfying the condtions of the main body of the problem, and , satisfying the condition of Statement 1. Since the triangles are congruent, all three pairs of corresponding sides have the same length, so the perimeters are equal.
Case 2: Examine this diagram, which superimposes the triangles such that 
and 
coincide with 
and 
, respectively:
The conditions of the main body and Statement 1 are met, since 
, 
(their being the same segment and angle, respectively, in the diagram), and 
by construction. Note, however, that 
, so:
.
Making the perimeters different.
Now assume Statement 2 alone. 
is the included side of 
and 
, and 
is the included side of 
and 
. The three congruence statements given in the main body and Statement 2 together set the conditions of the Angle-Side-Angle Postulate, so 
, and the perimeters are indeed the same.
Assume Statement 1 alone. We show that this provides insufficient information by examining two scenarios.
Case 1:
Case 2: Examine this diagram, which superimposes the triangles such that
The conditions of the main body and Statement 1 are met, since
Making the perimeters different.
Now assume Statement 2 alone.
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What is the perimeter of ?
Statement 1: 
and 
Statement 2: 
What is the perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
From Statement 1 alone, we know that two sides are of length 10, but we are not given any clue as to the length of the third. Therefore, we cannot calculate the sum of the side lengths, which is the perimeter. Statement 2 alone is unhelpful, since it only gives that two angles have equal measure; applying the Converse of the Isosceles Triangle Theorem, it can be determined that their opposite sides 
and 
are congruent, but no actual side lengths can be found.
Now assume both statements to be true. From Statement 1, 
, and, as stated before, it follows from Statement 2 that 
. Therefore, the triangle is equilateral with sides of common length 10, making the perimeter 30.
From Statement 1 alone, we know that two sides are of length 10, but we are not given any clue as to the length of the third. Therefore, we cannot calculate the sum of the side lengths, which is the perimeter. Statement 2 alone is unhelpful, since it only gives that two angles have equal measure; applying the Converse of the Isosceles Triangle Theorem, it can be determined that their opposite sides
Now assume both statements to be true. From Statement 1,
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Find the perimeter of the obtuse .
I) .
II) .
Find the perimeter of the obtuse
I)
II)
Tap to reveal answer
We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.
I) Relates the two shorter sides.
II) Relates the longest side to one of the short sides.
However, we cannot find any of our side lengths, so we cannot find the perimeter.
We are told PGN is obtuse, so it has one angle larger than 90 degrees. However, we don't know what that angle is. To find the perimeter we need all three sides.
I) Relates the two shorter sides.
II) Relates the longest side to one of the short sides.
However, we cannot find any of our side lengths, so we cannot find the perimeter.
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Give the perimeter of .
Statement 1:
Statement 2:
Give the perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
We demonstrate that both statements together provide insufficient information by examining two cases:
Case 1:
The perimeter of is
.
Case 2: .
The perimeter of is
.
Both cases satisfy the conditions of both statements but different perimeters are yielded.
We demonstrate that both statements together provide insufficient information by examining two cases:
Case 1:
The perimeter of
Case 2:
The perimeter of
Both cases satisfy the conditions of both statements but different perimeters are yielded.
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True or false: The perimeter of is greater than 24.
Statement 1:
Statement 2:
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:
Case 1:
This triangle satisfies the triangle inequality, since ; its perimeter is
Case 2:
This triangle satisfies the triangle inequality, since ; its perimeter is .
Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.
Assume Statement 2 alone. Again, ; since, by Statement 2, , by substitution, . The perimeter of is
, and, since , then
The perimeter of is greater than 24.
Statement 1 alone gives insufficient information. By the Triangle Inequality Theorem, the sum of the lengths of the shortest two sides of a triangle must be greater than the length of the longest. Examine these two scenarios:
Case 1:
This triangle satisfies the triangle inequality, since
Case 2:
This triangle satisfies the triangle inequality, since
Therefore, Statement 1 alone does not answer whether the perimeter is less than, equal to, or greater than 24.
Assume Statement 2 alone. Again,
The perimeter of
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True or false: The perimeter of is greater than 50.
Statement 1: is an isosceles triangle.
Statement 2: and
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
Also,
Therefore,
,
and we can find the range of the values of the perimeter
by adding:
Therefore, the perimeter may or may not be greater than 50.
Assume both statements to be true. An isosceles triangle has two sides of the same length, so either or .
If , the perimeter is
If , the perimeter is
Either way, the perimeter is less than 50.
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
Also,
Therefore,
and we can find the range of the values of the perimeter
by adding:
Therefore, the perimeter may or may not be greater than 50.
Assume both statements to be true. An isosceles triangle has two sides of the same length, so either
If
If
Either way, the perimeter is less than 50.
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Give the perimeter of .
Statement 1:
Statement 2:
Give the perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
The perimeter of is equal to the sum of the lengths of the sides; that is, .
From Statement 1 alone, we get
we can add to both sides to get
However, without any further information, we cannot determine the actual perimeter.
A similar argument shows that Statement 2 alone gives insufficient information as well.
However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:
Divide both sides by 3:
Since
,
we can substitute 29 for and find :
While we cannot find or individually, this is not necessary; in the perimeter formula, we can substitute 29 for and 8 for :
.
The perimeter of
From Statement 1 alone, we get
we can add
However, without any further information, we cannot determine the actual perimeter.
A similar argument shows that Statement 2 alone gives insufficient information as well.
However, suppose we were to multiply both sides of the equation in Statement 1 by 2, then add both sides of Statement 2:
Divide both sides by 3:
Since
we can substitute 29 for
While we cannot find
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True or false: The perimeter of is greater than 50.
Statement 1: is an isosceles triangle.
Statement 2: and
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
We can find the minimum of the perimeter:
:
The perimeter of is greater than 50.
Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.
Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so
We can find the minimum of the perimeter:
The perimeter of
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True or false: The perimeter of is greater than 60.
Statement 1: is an isosceles triangle.
Statement 2: and
True or false: The perimeter of
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements to be true. An isosceles triangle has two sides of equal length, so either or . By the Triangle Inequality Theorem, the length of each side must be less than the sum of the lengths of the other two; both scenarios are possible, since
and
.
If , the perimeter of is
.
If , the perimeter of is
.
Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.
Assume both statements to be true. An isosceles triangle has two sides of equal length, so either
and
If
If
Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.
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