Triangles - GMAT Quantitative
Card 0 of 1496
Find the hypotenuse of an obtuse triangle.
Statement 1: Two given lengths with an inscribed angle.
Statement 2: Two known angles.
Find the hypotenuse of an obtuse triangle.
Statement 1: Two given lengths with an inscribed angle.
Statement 2: Two known angles.
Statement 1: Two given lengths with an inscribed angle.
Draw a picture of the scenario. The values of 
, 
, and angle 
are known values.
Use the Law of Cosines to determine side length 
.
Statement 2: Two known angles.
There is insufficient information to solve for the length of the hypotenuse with only two interior angles. The third angle can be determined by subtracting the 2 angles from 180 degrees.
The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles. There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines.
Therefore:
Statement 1: Two given lengths with an inscribed angle.
Draw a picture of the scenario. The values of
Use the Law of Cosines to determine side length
Statement 2: Two known angles.
There is insufficient information to solve for the length of the hypotenuse with only two interior angles. The third angle can be determined by subtracting the 2 angles from 180 degrees.
The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles. There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines.
Therefore:
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For obtuse triangle ABC, what is the length of c?
(1) 
and 
(2) c is an integer, 
For obtuse triangle ABC, what is the length of c?
(1)
(2) c is an integer,
Since this is an obtuse triangle, pythagorean theorem does not apply.
Statement 1 by itself will only determine a range of values c utilizing the 3rd side rule of triangles. 
. Therefore, statement 1 alone is insufficient.
Statement 2 by itself will determine that c is either 10 or 11. Therefore, statement 2 alone is insufficient.
When taken together, statements 1 and 2 define a definitive value for c: 
. Therefore, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
Since this is an obtuse triangle, pythagorean theorem does not apply.
Statement 1 by itself will only determine a range of values c utilizing the 3rd side rule of triangles.
Statement 2 by itself will determine that c is either 10 or 11. Therefore, statement 2 alone is insufficient.
When taken together, statements 1 and 2 define a definitive value for c:
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. What is the measure of c?
(1) 
(2) 
(1)
(2)
Since 
, 
, 
, therefore, 
thus making this an acute triangle. Pythagorean theorem will not apply.
With the information in statement 1, we can't determine the lengths of any other sides. Therefore, Statement 1 alone is not sufficient.
With the information in statement 2, we can't determine the lengths of any other sides. Therefore, Statement 2 alone is not sufficient.
Using the Third Side Rule for triangles, the information in statements 1 and 2 together would allow us to determine the range of values for c. 
, but this does not provide a definitive value for c. Therefore, Both statements together are not sufficient.
Therefore - the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.
Since
With the information in statement 1, we can't determine the lengths of any other sides. Therefore, Statement 1 alone is not sufficient.
With the information in statement 2, we can't determine the lengths of any other sides. Therefore, Statement 2 alone is not sufficient.
Using the Third Side Rule for triangles, the information in statements 1 and 2 together would allow us to determine the range of values for c.
Therefore - the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.
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Find the length of the hypotenuse of obtuse triangle TLC:
I) 
II) Side T is 
Find the length of the hypotenuse of obtuse triangle TLC:
I)
II) Side T is
Find the length of the hypotenuse of obtuse triangle TLC:
I) 
II) Side T is 
Using I), we can find the measure of all 3 angles:
Next, use II) and the Law of sines to find the hypotenuse:
And we needed both statements to find it!
Find the length of the hypotenuse of obtuse triangle TLC:
I)
II) Side T is
Using I), we can find the measure of all 3 angles:
Next, use II) and the Law of sines to find the hypotenuse:
And we needed both statements to find it!
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Consider equilateral triangle 
.
I) The area of triangle 
is 
.
II) Side 
is 
.
What is the height of 
?
Consider equilateral triangle
I) The area of triangle
II) Side
What is the height of
Since is states that we are working with a equilateral triangle we can use the formula for area:
where 
is the side length. Once we have calculated the side length we can then plug that value along with the area into the equation:
and solve for h.
Consider that equilateral triangles have equal sides. This means we can make ABC into two smaller triangles with hypotenuse of 13 and base of 6.5. We can use that to find the height. We can find the height using statement II.
Therefore, both statements alone are sufficient to solve the question.
Since is states that we are working with a equilateral triangle we can use the formula for area:
Consider that equilateral triangles have equal sides. This means we can make ABC into two smaller triangles with hypotenuse of 13 and base of 6.5. We can use that to find the height. We can find the height using statement II.
Therefore, both statements alone are sufficient to solve the question.
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What is the length of the height of 
?
(1) 
, 
is the midpoint of 
(2) 
What is the length of the height of
(1)
(2)
Firslty, we would need to have the length of the other sides of the triangles to calculate the height. Information about the angles could also be able to see whether the triangle is a special triangle.
From statement one we can say that triangle ABC is an equilateral triangle, since D is the mid point of the the basis. Moreover knowing 
, we can see that angle 
is 60 degrees. Since D, the basis of the height is the midpoint it follows that 
is also 60 degrees. Therefore 
is also 60 degrees. Hence the triangle is equilateral. However, we don't know the length of any of the side.
Statement 2 gives us the piece of missing information. And alone statement 2 doesn't help us find the height.
It follows that both statements together are sufficient.
Firslty, we would need to have the length of the other sides of the triangles to calculate the height. Information about the angles could also be able to see whether the triangle is a special triangle.
From statement one we can say that triangle ABC is an equilateral triangle, since D is the mid point of the the basis. Moreover knowing
Statement 2 gives us the piece of missing information. And alone statement 2 doesn't help us find the height.
It follows that both statements together are sufficient.
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The equilateral triangle is inscribed in the circle. What is the length of the height?
(1) The center of the circle is at 
of the vertices A, B and C.
(2) 
.
The equilateral triangle is inscribed in the circle. What is the length of the height?
(1) The center of the circle is at
(2)
To be able to answer the question, we would need information about the radius or about the sides of the triangle.
Statement 1 tells us that the center of the circle is at 
of the vertice. However this is a property and it will be the same in any equilateral triangle inscribed in a circle, indeed, the heights, whose intersection is the center of gravity, all intersect at 
of the vertices.
Statement 2 also tells us something that we could have known from the properties of equilateral triangles. Indeed, equilateral triangles have all their 3 angles equal to 
.
Even by taking both statements together, we can't tell anything about the lengths of the height. Therefore the statements are insufficient.
To be able to answer the question, we would need information about the radius or about the sides of the triangle.
Statement 1 tells us that the center of the circle is at
Statement 2 also tells us something that we could have known from the properties of equilateral triangles. Indeed, equilateral triangles have all their 3 angles equal to
Even by taking both statements together, we can't tell anything about the lengths of the height. Therefore the statements are insufficient.
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Consider the equilateral 
.
I) Side 
.
II) 
has an area of 
.
What is the height of 
?
Consider the equilateral
I) Side
II)
What is the height of
I) Gives us the length of side w. Since this is an equilateral triangle, we really are given all three sides. From here we can break WHY into two smaller triangles and use either Pythagorean Theorem (or 30/60/90 triangle ratios) to find the height.
II) Gives us the area of WHY. If we recognize the fact that we can make two smaller 30/60/90 triangles from WHY, then we can make an equation with one variable to find the height.
Solve the following for b:
Thus, either statement is sufficient to answer the question.
I) Gives us the length of side w. Since this is an equilateral triangle, we really are given all three sides. From here we can break WHY into two smaller triangles and use either Pythagorean Theorem (or 30/60/90 triangle ratios) to find the height.
II) Gives us the area of WHY. If we recognize the fact that we can make two smaller 30/60/90 triangles from WHY, then we can make an equation with one variable to find the height.
Solve the following for b:
Thus, either statement is sufficient to answer the question.
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Given equilateral triangles 
and 
, construct the altitude from 
to 
on 
, and the altitude from 
to 
on 
.
Which, if either, of 
and 
is longer?
Statement 1: 
Statement 2: 
Given equilateral triangles
Which, if either, of
Statement 1:
Statement 2:
Let 
and 
be the common side lengths of 
and 
. The length of an altitude of a triangle is solely a function of its side length, so it follows that the triangle with the greater side length is the one whose altitude is the longer. Therefore, the question is equivalent to which, if either, of 
or 
is the greater.
Assume Statement 1 alone. This statement can be rewritten as
It follows that 
has the greater side length, and, consequently, that its altitude 
is longer than 
.
Assume Statement 2 alone. 
divides the triangle into two congruent triangles, so 
is the midpoint of 
; therefore, 
. Statement 2 can be rewritten as
This statement is inconclusive. Suppose 
—that is, each side of 
is of length 1. Then 
, 
, and 
all make that inequality true; without further information, it is therefore unclear whether 
, the side length of 
, is less than, equal to, or greater than 
, the side length of 
. Consequently, it is not clear which triangle has the longer altitude.
Let
Assume Statement 1 alone. This statement can be rewritten as
It follows that
Assume Statement 2 alone.
This statement is inconclusive. Suppose
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Given equilateral triangles 
and 
, construct the altitude from 
to 
on 
, and the altitude from 
to 
on 
.
True or false: 
or 
have the same length.
Statement 1: 
and 
are chords of the same circle.
Statement 2: 
and 
have the same area.
Given equilateral triangles
True or false:
Statement 1:
Statement 2:
Statement 1 alone is inconclusive, since chords of the same circle can have different lengths.
Statement 2 alone is conclusive. The common side length 
of an equilateral triangle depends solely on the area, so it follows that the sides of two triangles of equal area will have the same common side length. Also, each altitude divides its triangle into two 30-60-90 triangles. Examining 
and 
, we can easily find that these triangles are congruent by way of the Angle-Side-Angle. Postulate, so it follows by triangle congruence that 
.
Statement 1 alone is inconclusive, since chords of the same circle can have different lengths.
Statement 2 alone is conclusive. The common side length
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Given equilateral triangles 
and 
, construct the altitude from 
to 
on 
, and the altitude from 
to 
on 
.
Which, if either, is longer, 
or 
?
Statement 1: 
Statement 2: 
Given equilateral triangles
Which, if either, is longer,
Statement 1:
Statement 2:
Assume Statement 1 alone. If altitude 
of 
is constructed, the right triangle 
is constructed as a consequence. 
is a leg and 
the hypotenuse of 
, so 
. Since by Statement 1, it is given that 
, then by substitution, 
, so 
is the longer altitude.
Assume Statement 2 alone.
, so
divides 
into two 30-60-90 triangles, one of which is 
with shorter leg 
and hypotenuse 
, so by the 30-60-90 Theorem,
Again, 
and 
is the longer altitude.
Assume Statement 1 alone. If altitude
Assume Statement 2 alone.
Again,
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is an equilateral triangle. An altitude of 
is constructed from 
to a point 
on 
.
What is the length of 
?
Statement 1: 
is inscribed inside a circle of circumference 
.
Statement 2: 
is a chord of a circle of area 
.
What is the length of
Statement 1:
Statement 2:
Assume Statement 1 alone. The circumscribed circle, or "circumcircle," of a triangle has as its center the mutual point of intersection of the perpendicular bisectors of the three sides, which, in the case of an equilateral triangle, coincide with the altitudes. 
, its three altitudes, and the circumcircle are shown below:
The circle has circumference 
, so its radius, which is equal to the length of 
, can be found by dividing this by 
to yield
.
Also, the point of intersection of the three altitudes divides each altitude into two segments, the ratio of whose lengths is 2 to 1, so 
.
Assume Statement 2 alone. The radius of the circle can be found using the area formula for the circle, and the diameter can be found by doubling this. This diameter, however, only provides an upper bound for the length of a chord of the circle; if 
is a chord of this circle, its length cannot be determined, only a range in which its length must fall. Therefore, Statement 2 is insufficient.
Assume Statement 1 alone. The circumscribed circle, or "circumcircle," of a triangle has as its center the mutual point of intersection of the perpendicular bisectors of the three sides, which, in the case of an equilateral triangle, coincide with the altitudes.
The circle has circumference
Also, the point of intersection of the three altitudes divides each altitude into two segments, the ratio of whose lengths is 2 to 1, so
Assume Statement 2 alone. The radius of the circle can be found using the area formula for the circle, and the diameter can be found by doubling this. This diameter, however, only provides an upper bound for the length of a chord of the circle; if
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is an equilateral triangle. An altitude of 
is constructed from 
to a point 
on 
.
What is the length of 
?
Statement 1: 
has perimeter 36.
Statement 2: 
has area 
.
What is the length of
Statement 1:
Statement 2:
From either statement alone, it is possible to find the length of one side of 
; from Statement 1 alone, the perimeter 36 can be divided by 3 to yield side length 12, and from Statement 2 alone, the area formula for an equilateral triangle can be applied as follows:
Once this is found, the length of altitude 
can be found by noting that this divides the triangle into two congruent 30-60-90 triangles and by applying the 30-60-90 Theorem:
and
From either statement alone, it is possible to find the length of one side of
Once this is found, the length of altitude
and
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is an equilateral triangle. An altitude of 
is constructed from 
to a point 
on 
.
True or false: 
Statement 1: A circle of area less than 
can be inscribed inside 
.
Statement 2: 
is a chord of a circle of area 
.
True or false:
Statement 1: A circle of area less than
Statement 2:
Assume Statement 1 alone. The inscribed circle, or "incircle_,_" of a triangle has as its center the mutual point of intersection of the bisectors of the three angles, which, in the case of an equilateral triangle, coincide with the altitudes. 
, its three altitudes, and the incircle are shown below:
If the area of the incircle is less than 
, then the upper bound of the radius, which is 
, can be found as follows:
and 
has length less than 4. Also, the point of intersection of the three altitudes divides each altitude into two segments, the ratio of whose lengths is 2 to 1, so
and
Therefore, Statement 1 only tells us that 
, leaving open the possibility that 
may be less than, equal to or greater than 10.
Assume Statement 2 alone. The radius of a circle of area 
can be found as follows:
The diameter of the circle is twice this, or 
. Since the longest chords of a circle are its diameters, then any chord in this circle must have length less than or equal to this. Statement 2 tells us that
Now examine the above diagram. 
, as half of an equilateral triangle, is a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,
and
is therefore a true statement.
Assume Statement 1 alone. The inscribed circle, or "incircle_,_" of a triangle has as its center the mutual point of intersection of the bisectors of the three angles, which, in the case of an equilateral triangle, coincide with the altitudes.
If the area of the incircle is less than
and
and
Therefore, Statement 1 only tells us that
Assume Statement 2 alone. The radius of a circle of area
The diameter of the circle is twice this, or
Now examine the above diagram.
and
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Given 
and 
, with 
an equilateral triangle. Construct the altitude from 
to 
on 
, and the altitude from 
to 
on 
.
Which, if either, of and is longer?
Statement 1: 
Statement 2: 
is a right angle.
Given
Which, if either, of
Statement 1:
Statement 2:
Assume both statements are true. From Statement 1 alone, 
, and 
, so 
and 
. Therefore, between 
and 
, two pairs of corresponding sides are congruent.
is an equilateral triangle, so 
; from Statement 2, 
is a right angle, so 
. This means that the included angle in 
is of greater measure, so by the Side-Angle-Side Inequality Theorem, or Hinge Theorem, it has the longer opposite side, or 
. Both triangles are isosceles, so both altitudes divide the triangles into congruent right triangles, and by congruence, 
and 
are the midpoints of their respective sides. This means that
By the Pythagorean Theorem,
and
Since 
and 
,
meaning that is the longer altitude.
Note that this depended on knowing both statements to be true. Statement 1 alone is insufficient, since, for example, had 
measured less than 
, then by the same reasoning, would have been the shorter altitude. Statement 2 alone is insufficient because it gives information only about one angle, and nothing about any side lengths.
Assume both statements are true. From Statement 1 alone,
By the Pythagorean Theorem,
and
Since
meaning that
Note that this depended on knowing both statements to be true. Statement 1 alone is insufficient, since, for example, had
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What is the perimeter of isosceles triangle ABC?
(1) 
(2) 
What is the perimeter of isosceles triangle ABC?
(1)
(2)
(1) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
(2) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
Additionally, since we don't know which one of the sides (
or 
) is one of the equal sides, it's impossible to determine the perimeter given the information provided.
(1) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
(2) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
Additionally, since we don't know which one of the sides (
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Find the perimeter of right triangle 
.
I) 
II) 
Find the perimeter of right triangle
I)
II)
If the two shorter sides of a right triangle are equal, that means our other two angles are 45 degrees. This means our triangle follows the ratios for a 45/45/90 triangle, so we can find the remaining sides from the length of the hypotenuse.
I) Tells us we have a 45/45/90 triangle. The ratio of side lengths for a 45/45/90 triangle is 
.
II) Tells us the length of the hypotenuse.
Together, we can find the remaining two sides and then the perimeter.
If the two shorter sides of a right triangle are equal, that means our other two angles are 45 degrees. This means our triangle follows the ratios for a 45/45/90 triangle, so we can find the remaining sides from the length of the hypotenuse.
I) Tells us we have a 45/45/90 triangle. The ratio of side lengths for a 45/45/90 triangle is
II) Tells us the length of the hypotenuse.
Together, we can find the remaining two sides and then the perimeter.
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Find the perimeter of the right triangle.
- The product of the base and height measures
.
- The hypotenuse measures
.
Find the perimeter of the right triangle.
- The product of the base and height measures
- The hypotenuse measures
Statement 1: We need additional information.
But this can mean our base and height measure 2 and 24, 4 and 12, or 6 and 8.
We cannot determine which one based solely on this statement.
Statement 2: We're given the length of the hypotenuse so we can narrow down the possible base and height values.
We have to see which pair of values makes the statement 
true.
The only pair that does is 6 and 8.
We can now find the perimeter of the right triangle:
or, if you're more familiar with the equation 
, then:
Statement 1: We need additional information.
But this can mean our base and height measure 2 and 24, 4 and 12, or 6 and 8.
We cannot determine which one based solely on this statement.
Statement 2: We're given the length of the hypotenuse so we can narrow down the possible base and height values.
We have to see which pair of values makes the statement
The only pair that does is 6 and 8.
We can now find the perimeter of the right triangle:
or, if you're more familiar with the equation
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Calculate the perimeter of the triangle.
- The hypotenuse of the right triangle is
.
- The legs of the right triangle measure
and 
.
Calculate the perimeter of the triangle.
- The hypotenuse of the right triangle is
- The legs of the right triangle measure
Statement 1: In order to find the perimeter of a right triangle, we need to know the lengths of the legs, not the hypotenuse.
Statement 2: Since we have the values to both of the legs' lengths, we can just plug it into the equation for the perimeter:
Statement 1: In order to find the perimeter of a right triangle, we need to know the lengths of the legs, not the hypotenuse.
Statement 2: Since we have the values to both of the legs' lengths, we can just plug it into the equation for the perimeter:
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Consider equilateral triangle .
I) The area of triangle is .
II) Side is .
What is the height of ?
Consider equilateral triangle
I) The area of triangle
II) Side
What is the height of
Since is states that we are working with a equilateral triangle we can use the formula for area:
where is the side length. Once we have calculated the side length we can then plug that value along with the area into the equation:
and solve for h.
Consider that equilateral triangles have equal sides. This means we can make ABC into two smaller triangles with hypotenuse of 13 and base of 6.5. We can use that to find the height. We can find the height using statement II.
Therefore, both statements alone are sufficient to solve the question.
Since is states that we are working with a equilateral triangle we can use the formula for area:
Consider that equilateral triangles have equal sides. This means we can make ABC into two smaller triangles with hypotenuse of 13 and base of 6.5. We can use that to find the height. We can find the height using statement II.
Therefore, both statements alone are sufficient to solve the question.
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