GMAT Math : DSQ: Calculating the length of the side of an equilateral triangle

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Example Question #11 : Equilateral Triangles

Given equilateral triangle  and right triangle , which, if either, is longer,   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 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:

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

Explanation:

Assume Statement 1 alone. Since all three sides of  are congruent - specifically,  - and , it follows by transitivity that . However, no information is given as to whether  has length greater than, equal to, or less than , so it cannot be determined which of  and , if either, is the longer. By a similar argument, Statement 2 yields insufficient information.

Now assume both statements are true.  and  are each congruent to one of the congruent sides of equilateral  and are therefore congruent to each other. However, the hypotenuse of a right triangle must be longer than both legs, so the hypotenuse of   is  is also longer than any segment congruent to one of the legs, which includes all three sides of  - specificially,  is longer than .

Example Question #12 : Equilateral Triangles

 is equilateral.  may or may not be equilateral. 

which, if either, is longer,   or  ?

Statement 1: 

Statement 2:  and 

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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.

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.  is equilateral, so . Also, by the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third, so . From Statement 1, , so by substitution, , and .

Statement 2 alone provides insufficient information. For example, assume  is an equilateral triangle with sidelength 9. If  is an equilateral triangle with sidelength 8, the conditions of the statement hold, and . However, if  is a right triangle in which , and , the conditions of the statement still hold, but .

Example Question #13 : Equilateral Triangles

Given equilateral triangles  and , which, if either, is longer,   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 insufficient 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.

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:

All sides of an equilateral triangle have the same measure, so we can let  be the common sidelength of , and  be that of .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater,  or . The two statements together are insufficient to answer the question, however; 5 and 10 have sum 15 and product 50, but we cannot determine without further information whether   and , or vice versa. Therefore, we do not know for sure whether a side of  is longer than a side of  - specifically, which of  or  is longer.

Example Question #14 : Equilateral Triangles

Given equilateral triangles  and , 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.

EITHER statement ALONE is sufficient to answer the question.

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.

Correct answer:

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

Explanation:

All sides of an equilateral triangle have the same measure, so we can let  be the common sidelength of , and  be that of .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater,  or .

Statement 1 alone yields insufficient information; for example, the two numbers added together could be 10 and 14, but it is impossible to determine whether  or  is the greater of the two. Statement 2 alone is also insufficient, for a similar reason; for example, the two numbers could be 9 and 16, but again, either  or  could be the greater.

Now assume both statements. The only two numbers that can be added to yield a sum of 24 and multiplied to yield a product of 144 are 12 and 12; therefore, , and  and  have the same sidelengths. Specifically,  and  have the same length.

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