# GMAT Math : Equilateral Triangles

## Example Questions

### Example Question #31 : Equilateral Triangles

Consider the equilateral .

I) Side .

II)   has an area of .

What is the height of ?

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Explanation:

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.

### Example Question #5 : Dsq: Calculating The Height Of An Equilateral Triangle

is an equilateral triangle. An altitude of  is constructed from  to a point  on .

What is the length of ?

Statement 1:  has perimeter 36.

Statement 2:  has area .

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

### Example Question #6 : Dsq: Calculating The Height Of An Equilateral Triangle

Given equilateral triangles  and , construct the altitude from  to  on , and the altitude from  to  on .

Which, if either, is longer,  or ?

Statement 1:

Statement 2:

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.

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.

Explanation:

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.

### Example Question #7 : Dsq: Calculating The Height Of An Equilateral Triangle

is an equilateral triangle. An altitude of  is constructed from  to a point  on .

True or false:

Statement 1: A circle of area less than  can be inscribed inside

Statement 2:  is a chord of a circle of area .

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

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

Explanation:

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.

### Example Question #8 : Dsq: Calculating The Height Of An Equilateral Triangle

is an equilateral triangle. An altitude of  is constructed from  to a point  on .

What is the length of  ?

Statement 1:  is inscribed inside a circle of circumference .

Statement 2:  is a chord of a circle of area .

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

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 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.

Explanation:

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.

### Example Question #9 : Dsq: Calculating The Height Of An Equilateral Triangle

Given  and , with  an equilateral triangle. Construct the altitude from  to  on , and the altitude from  to  on .

Which, if either, of   and  is longer?

Statement 1:

Statement 2:  is a right angle.

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 insufficient 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

### Example Question #10 : Dsq: Calculating The Height Of An Equilateral Triangle

Given equilateral triangles  and , construct the altitude from  to  on , and the altitude from  to  on .

True or false:  or  have the same length.

Statement 1:  and  are chords of the same circle.

Statement 2:   and  have the same area.

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.

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

Explanation:

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 .

### Example Question #31 : Equilateral Triangles

Given equilateral triangles  and , construct the altitude from  to  on , and the altitude from  to  on .

Which, if either, of   and  is longer?

Statement 1:

Statement 2:

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.

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

Explanation:

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.

### Example Question #1 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

What is the perimeter of

(1) The area of the triangle is .

(2)  is an equilateral triangle.

Each statement alone is sufficient

Statement 1 alone is sufficient

Both statements together are sufficient

Statement 2 alone is sufficient

Statements 1 and 2 taken together are not sufficient

Both statements together are sufficient

Explanation:

To find the perimeter we should be able to calculate each sides of the triangle.

Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.

Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.

Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula: . Where  is the area and  the length of the side.

Therefore both statements are sufficient.

### Example Question #1 : Dsq: Calculating The Perimeter Of An Equilateral Triangle

Find the perimeter of  given the following:

I) .

II) Side .

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Both statements are needed to answer the question.

Explanation:

To find perimeter, we need the side lengths.

I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.

II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length.

Add up all the sides to get the perimeter.

We need I) and II) to find the perimeter