DSQ: Calculating the length of the side of an acute / obtuse triangle - GMAT Quantitative
Card 0 of 64
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
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For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
Note: Figure NOT drawn to scale.
The above shows a triangle 
inscribed inside a rectangle 
. is 
isosceles?
Statement 1: 
is the midpoint of 
.
Statement 2: 
Note: Figure NOT drawn to scale.
The above shows a triangle
Statement 1:
Statement 2:
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We show Statement 1 alone is sufficient:
If 
is the midpoint of 
, then 
. Opposite sides of a rectangle are congruent, so 
; all angles of a rectangle, being right angles, are congruent, so 
. This sets up the conditions for the Side-Angle-Side Theorem, and 
. Consequently, 
, and 
is isosceles.
Now, we show Statement 2 alone is sufficient:
If 
, and 
are congruent, then 
and 
, being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, 
is isosceles.
We show Statement 1 alone is sufficient:
If
Now, we show Statement 2 alone is sufficient:
If
Is 
isosceles?
Statement 1: 
Statement 2: 
Is
Statement 1:
Statement 2:
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Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of 
; Statement 2 only gives us information about another triangle.
Suppose we assume both statements. Then by similarity,
.
Since 
, then
, or
.
This makes 
isosceles.
Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of
Suppose we assume both statements. Then by similarity,
Since
This makes
Which side of 
is the longest?
Statement 1: 
is an obtuse angle.
Statement 2: 
and 
are both acute angles.
Which side of
Statement 1:
Statement 2:
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If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that 
is an obtuse angle, however, we can deduce that 
and 
are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that 
has the greatest measure, and that its opposite side, 
, is the longest.
If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that
Which of the three sides of 
is the longest?
Statement 1: 
Statement 2: 
Which of the three sides of
Statement 1:
Statement 2:
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The longest side of a triangle is opposite the angle of greatest measure.
From Statement 1 alone, we can find two possible scenarios with different answers:
Case 1:
Case 2:
In both cases, 
, but in Case 1, 
is the longest side, and in Case 2, 
is the longest side.
From Statement 2 alone, however, we know that 
, so 
is obtuse and the other two angles are acute. That makes 
the longest side.
The longest side of a triangle is opposite the angle of greatest measure.
From Statement 1 alone, we can find two possible scenarios with different answers:
Case 1:
Case 2:
In both cases,
From Statement 2 alone, however, we know that
True or false: 
is scalene.
Statement 1: 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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By definition, a scalene triangle has three noncongruent sides.
If 
, then and , and the triangle is scalene.
If 
, then and , but 
, so the triangle is not scalene.
The two statements together are insufficient.
By definition, a scalene triangle has three noncongruent sides.
If
If
The two statements together are insufficient.
True or false: 
is scalene.
Statement 1: 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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Assume both statements are true.
By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, 
. Statement 2 alone establishes that 
. However, the two statements together do not establish whether or not 
, so it is not clear whether 
is scalene or isosceles.
Assume both statements are true.
By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1,
True or false: 
is scalene.
Statement 1: 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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By definition, a scalene triangle has three noncongruent sides.
Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side 
is congruent to either of the other sides.
Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that 
. The triangle cannot be scalene.
By definition, a scalene triangle has three noncongruent sides.
Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side
Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
Tap to see back →
For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
Note: Figure NOT drawn to scale.
The above shows a triangle inscribed inside a rectangle . is isosceles?
Statement 1: is the midpoint of .
Statement 2:
Note: Figure NOT drawn to scale.
The above shows a triangle
Statement 1:
Statement 2:
Tap to see back →
We show Statement 1 alone is sufficient:
If is the midpoint of , then . Opposite sides of a rectangle are congruent, so ; all angles of a rectangle, being right angles, are congruent, so . This sets up the conditions for the Side-Angle-Side Theorem, and . Consequently, , and is isosceles.
Now, we show Statement 2 alone is sufficient:
If , and are congruent, then and , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, is isosceles.
We show Statement 1 alone is sufficient:
If
Now, we show Statement 2 alone is sufficient:
If
Is isosceles?
Statement 1:
Statement 2:
Is
Statement 1:
Statement 2:
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Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of ; Statement 2 only gives us information about another triangle.
Suppose we assume both statements. Then by similarity,
.
Since , then
, or
.
This makes isosceles.
Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of
Suppose we assume both statements. Then by similarity,
Since
This makes
Which side of is the longest?
Statement 1: is an obtuse angle.
Statement 2: and are both acute angles.
Which side of
Statement 1:
Statement 2:
Tap to see back →
If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that is an obtuse angle, however, we can deduce that and are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that has the greatest measure, and that its opposite side, , is the longest.
If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that
Which of the three sides of is the longest?
Statement 1:
Statement 2:
Which of the three sides of
Statement 1:
Statement 2:
Tap to see back →
The longest side of a triangle is opposite the angle of greatest measure.
From Statement 1 alone, we can find two possible scenarios with different answers:
Case 1:
Case 2:
In both cases, , but in Case 1, is the longest side, and in Case 2, is the longest side.
From Statement 2 alone, however, we know that , so is obtuse and the other two angles are acute. That makes the longest side.
The longest side of a triangle is opposite the angle of greatest measure.
From Statement 1 alone, we can find two possible scenarios with different answers:
Case 1:
Case 2:
In both cases,
From Statement 2 alone, however, we know that
True or false: is scalene.
Statement 1:
Statement 2:
True or false:
Statement 1:
Statement 2:
Tap to see back →
By definition, a scalene triangle has three noncongruent sides.
If , then and , and the triangle is scalene.
If , then and , but , so the triangle is not scalene.
The two statements together are insufficient.
By definition, a scalene triangle has three noncongruent sides.
If
If
The two statements together are insufficient.
True or false: is scalene.
Statement 1:
Statement 2:
True or false:
Statement 1:
Statement 2:
Tap to see back →
Assume both statements are true.
By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, . Statement 2 alone establishes that . However, the two statements together do not establish whether or not , so it is not clear whether is scalene or isosceles.
Assume both statements are true.
By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1,
True or false: is scalene.
Statement 1:
Statement 2:
True or false:
Statement 1:
Statement 2:
Tap to see back →
By definition, a scalene triangle has three noncongruent sides.
Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side is congruent to either of the other sides.
Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that . The triangle cannot be scalene.
By definition, a scalene triangle has three noncongruent sides.
Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side
Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
Is the triangle isosceles?
Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).
Statement 2: $\overline{AB}$ = $\sqrt{18}$, $\overline{AC}$ = $\sqrt{17}$, $\overline{BC}$ = $\sqrt{17}$
Tap to see back →
For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.
Note: Figure NOT drawn to scale.
The above shows a triangle inscribed inside a rectangle . is isosceles?
Statement 1: is the midpoint of .
Statement 2:
Note: Figure NOT drawn to scale.
The above shows a triangle
Statement 1:
Statement 2:
Tap to see back →
We show Statement 1 alone is sufficient:
If is the midpoint of , then . Opposite sides of a rectangle are congruent, so ; all angles of a rectangle, being right angles, are congruent, so . This sets up the conditions for the Side-Angle-Side Theorem, and . Consequently, , and is isosceles.
Now, we show Statement 2 alone is sufficient:
If , and are congruent, then and , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, is isosceles.
We show Statement 1 alone is sufficient:
If
Now, we show Statement 2 alone is sufficient:
If
Is isosceles?
Statement 1:
Statement 2:
Is
Statement 1:
Statement 2:
Tap to see back →
Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of ; Statement 2 only gives us information about another triangle.
Suppose we assume both statements. Then by similarity,
.
Since , then
, or
.
This makes isosceles.
Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of
Suppose we assume both statements. Then by similarity,
Since
This makes
Which side of is the longest?
Statement 1: is an obtuse angle.
Statement 2: and are both acute angles.
Which side of
Statement 1:
Statement 2:
Tap to see back →
If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that is an obtuse angle, however, we can deduce that and are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that has the greatest measure, and that its opposite side, , is the longest.
If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.
If we know that