DSQ: Calculating the length of the side of an equilateral triangle - GMAT Quantitative

The two statements together provide insufficient information. A triangle with sides , , and is equilateral and has perimeter ; A triangle with sides , , and is not equilateral and has perimeter .

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

To find the length of the side, we would need to know anything about the lengths in the circle or in the triangle.

From statement 1, we can find the radius of the circle, which allows us to calculate the height of the triangle, since the radius is of the height. And finally since the triangle is equilateral, we can also calculate the length of the sides from the height.

Therefore statement 1 is sufficient.

Statement 2 also gives us useful information, indeed the perimeter is simply three times the length of the sides.

Therefore the final answer is each statement alone is sufficient.

I) Tells us the perimeter of the triangle.

II) Tells us that FHT is an equilateral triangle.

Taking these statements together we are able to find the side length by dividing the perimeter from statement I, by 3 since all side lengths of an equilateral are the same by statement II.

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.

An equilateral triangle has three sides of equal length, so and .

Assume Statement 1 alone. Since , then, by substitution, .

Assume Statement 2 alone. Since , it follows that , and again by substitution, .

We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:

The three altitudes intersect at , which divides each altitude into two segments whose lengths have ratio 2:1. is the center of both the circumscribed circle, whose radius is , and the inscribed circle, whose radius is .

Therefore, from Statement 1 alone and the area formula for a circle, we can find from the area of the circumscribed circle:

From Statement 2 alone and the circumference formula for a cicle, we can find from the circumference of the inscribed circle:

By symmetry, is a 30-60-90 triangle, and either way, , and .

An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.

Assume Statement 1 alone. is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.

Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is , has length times this, or . This is also the length of .

Assume Statement 1 alone, and let be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length, and , so .

Assume Statement 2 alone, and let be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:

and ,

so

.

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.

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 .

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 .

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 then, equal to, or less than , so which of and , if either, is the longer cannot be answered.

Assume Statement 2 alone. Since is the right angle of , is the hypotenuse and this the longest side, so and . However, no comparisons with the sides of can be made.

Now assume both statements are true. as a consequence of Statement 1, and as a consequence of Statement 2, so .

If , then

.

This makes an equiangular triangle.

If , and is equiangular, then, since corresponding angles of similar triangles are congruent, has the same angle measures, and is itself equiangular.

From either statement, since all equiangular triangles are equilateral, we can draw this conclusion about .

The two statements together provide insufficient information. A triangle with sides , , and is equilateral and has perimeter ; A triangle with sides , , and is not equilateral and has perimeter .

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

To find the length of the side, we would need to know anything about the lengths in the circle or in the triangle.

From statement 1, we can find the radius of the circle, which allows us to calculate the height of the triangle, since the radius is of the height. And finally since the triangle is equilateral, we can also calculate the length of the sides from the height.

Therefore statement 1 is sufficient.

Statement 2 also gives us useful information, indeed the perimeter is simply three times the length of the sides.

Therefore the final answer is each statement alone is sufficient.

I) Tells us the perimeter of the triangle.

II) Tells us that FHT is an equilateral triangle.

Taking these statements together we are able to find the side length by dividing the perimeter from statement I, by 3 since all side lengths of an equilateral are the same by statement II.

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.