DSQ: Calculating the length of the hypotenuse of an acute / obtuse triangle - GMAT Quantitative Reasoning
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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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