GMAT Math : DSQ: Calculating the length of the side of a right triangle

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Example Question #1 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Data sufficiency question- do not actually solve the question

The lengths of two sides of a triangle are 6 and 8. What is the length of the third side?

1. The length of the longest side is 8.

2. The triangle contains a right angle.

Possible Answers:

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

 

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

 

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Each statement alone is sufficient to answer the question

 

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

 

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

Explanation:

Knowing that the triangle has a right angle indicates the question can be solved using the Pythagorean Theorem, however it is unclear which side is the hypotenuse. For example, if 8 is not the hypotenuse, the length of the third side is 10. If 8 is the hypotenuse, the length of the third side is 5.3.

Additionally, if you only know that 8 is the longest side, the length of the third side could be anything greater than 2 and less than 8. Therefore, having both pieces of data will allow you to solve the problem.

Example Question #2 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Which of the three sides of  has the greatest measure? 

Statement 1:  and  are complementary angles.

Statement 2:  is not an acute angle.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The side opposite the angle of greatest measure is the longest of the three, so if we can determine which angle is the longest, we can answer this question.

It follows from Statement 1 by definition that , so, since the measures of the three angles total , making  right and the other two acute. This proves  has the greatest measure of the three.

It follows from Statement 2 that  is either right or obtuse; therefore, . Subsequently, the other two angles are acute, so again,  has the greatest measure of the three.

From either statement alone, we can therefore identify  as the side of greatest measure.

Example Question #371 : Geometry

What is the base length of the right triangle?

  1. The width is four times the length.
  2. The area of the right triangle is .
Possible Answers:

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: All we're given is the equation for finding the width, , which we'll use in the next statement.

Statement 2: Using the information from statement 1, we can set up an equation and solve for the length.

Statement 2 alone would not have provided sufficient information because we would have ended up with

and would not have been able to determine what the the values were. 

 

Example Question #3 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Given  is a right triangle, which side is the hypotenuse - , or  ?

Statement 1: 

Statement 2: 

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The hypotenuse of a right triangle is its longest side.

Assume both statements are true. Then we know that , being shorter than either of the other sides, cannot be the hypotenuse, but without further information, we cannot tell which of the two other sides is longer than the other. Therefore, we cannot identify the hypotenuse for certain.

Example Question #4 : Dsq: Calculating The Length Of The Side Of A Right Triangle

 has right angle  has right angle . Which, if either, is longer,  or ?

Statement 1: 

Statement 2: 

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since  and  are the right angles of their respective triangles,  and , the segments opposite the right angles, are their hypotenuses, and, subsequently, their longest sides. Specifically, . Since, from Statement 1, , it follows that .

Assume Statement 2 alone. Again,  is the longest side of its triangle, so  . But we cannot determine whether  or  without further information.

Example Question #5 : Dsq: Calculating The Length Of The Side Of A Right Triangle

 is a right triangle. Evaluate .

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Neither statement alone gives enough information to find , as each alone gives only one sidelength.

Assume both statements are true. While neither side is indicated to be a leg or the hypotenuse, the hypotenuse of a right triangle is longer than either leg; therefore, since  and  are of equal length, they are the legs.  is the hypotenuse of an isosceles right triangle with legs of length 10, and by the 45-45-90 Theorem, the length of  is  times that of a leg, or .

Example Question #375 : Geometry

Given  is a right triangle, which side is the hypotenuse - , or  ?

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The side opposite of the right angle is the hypotenuse. Statement 1 alone eliminates  as the right angle, and, as a consequence,  as the hypotenuse - but only .

From Statement 2 alone, we have that , meaning that 

and 

Since  is shorter than , and only , is eliminated as the hypotenuse.

If both statements are assumed to be true, however, then both  and  can be eliminated as the hypotenuse, leaving  as the only choice.

Example Question #6 : Dsq: Calculating The Length Of The Side Of A Right Triangle

 has right angle  has right angle . Which, if either, is longer,  or ?

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

We are being asked to compare the lengths of the hypotenuses of the two triangles, since  and  are the sides opposite the right angles of their respective triangles.

Statement 1 alone gives insufficient information, as shown by examining these two cases.

Case 1: 

By the Pythagorean Theorem, the hypotenuse  has length

The hypotenuse  has length

Case 2: 

The hypotenuse  has length

and, as in Case 1,  has length .

In both cases,  and , so . But in the first case,  was longer than , and in the second case, the reverse was true.

Statement 2 is insufficient in that it only gives us the congruence of one set of corresponding legs; without further information, it is impossible to determine which hypotenuse is longer.

Now assume both statements are true. Since  and , by the subtraction property of inequality, 

and 

It follows from  and  that  and ; by the addition property of inequality, 

By the Pythagorean Theorem, 

and 

,

so the above inequality becomes, by substitution,

and 

,

proving that  is longer than .

Example Question #7 : Dsq: Calculating The Length Of The Side Of A Right Triangle

Given  is a right triangle, which side is the hypotenuse - , or  ?

Statement 1: 

Statement 2: 

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

In a right triangle, the angle of greatest measure is the right angle, and the side opposite it is the hypotenuse.

Assume both statements are true. We can eliminate  as the right angle, as it has measure less than both  and . However, we have no information that tells us which of   and  has the greater measure, so we cannot determine which is the right angle. Subsequently, we cannot eliminate either of their opposite sides,  or , respectively, as the hypotenuse.

Example Question #8 : Dsq: Calculating The Length Of The Side Of A Right Triangle

 has right angle  has right angle . Which, if either, is longer,  or ?

Statement 1: 

Statement 2: 

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The two statements together only give information about the angle measures of the two triangles. Without any information about the relative or absolute lengths of the sides, no comparison can be drawn between their hypotenuses.

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