Equilateral Triangles - GMAT Quantitative

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.

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.

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.

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.

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.

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 .

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. 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.

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

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 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.

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.

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.

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.

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.

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.

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 .

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. 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.

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