Equilateral Triangles - GMAT Quantitative Reasoning

To answer this question, we should be able to have information about any lengths in the circle or in the triangle, or also about the areas of the different regions.

Statement 1 alone is sufficient, since from the area of the circle we can find the length of the radius, which allows us to calculate the length of the height of the equilateral triangle. This would allow us to find the length of a side, giving us all the necessary information to calculate the area of the equilateral triangle.

Statement 2, although giving us information about the difference of the circle and the triangle, we cannot conclude anything because we don't know the proportions of the respective geometric figures. Therefore statement 2 is insufficient.

To conclude, statement one alone is sufficient.

To find the area of a triangle, we need the base and the height. Taking both statements together we can solve this problem.

I) tells us that OHT is an equilateral triangle.

II) gives us one side length. Which means we really know all the side lengths.

We can use either Pythagorean Theorem or 30/60/90 triangle ratios to find the height of OHT.

From there we can find the area.

Thus, both are needed. I) Tells us we have an equilateral, II) Lets us find the height. Both allow us to find the area.

In an equilateral triangle all sides are of the same length and all internal angles measure to .

Statement 1:

Where represents the length of the side.

If we're given the side, we can calculate the area:

Statement 2: We don't need the angle to find the area.

Assume Statement 1 alone. Since the vertices of both the triangle and the square are on the same circle, then the same circle is circumscribed about both polygons. For the sake of simplicity, we will assume the radius of that circle to be 1 (and the diameter to be 2); this argument will work regardless of the size of the circle.

A diagonal of a square doubles as a diameter of the circumscribed circle, so each diagonal of Square is 2; since a Square is a rhombus, the area is one half the product of the lengths of the diagonals, or .

Now examine the figure below, which shows , its altitudes, and its circumscribed circle:

The three altitudes of an equilateral triangle meet at the center of the circumscribed circle, or circumcenter, so ; they also divide one another into segments such that one has twice the length of the other, so . Therefore, . Also, the six smaller triangles are all 30-60-90 by symmetry, so by the 30-60-90 Theorem, , and .

Therefore, the area of the triangle is

,

which can be shown to be less than the square's area of 2.

Assume Statement 2 alone. Again, for simplicity's sake, we will use , so we can keep the area of the equilateral triangle as before, ; this argument works regardless of the dimensions. The length of a diagonal of the square will as before be , and its area will again be 2.

Assume Statement 1 alone. To find the diameter of a circle with circumference , divide the circumference by to get . This is also the length of each side of the triangle, so we can get the area using the area formula:

.

Assume Statement 2 alone. A 45-45-90 Triangle has congruent legs, and the area is half the product of their lengths, so if we let be the common sidelength,

By the 45-45-90 Theorem, the hypotenuse has length times this, or .

Since it is not given whether is a leg or the hypotenuse of a right triangle, however, the length of - and consequently, the area - is not clear.

Assume Statement 1 alone to be true. Since , it follows that the area of , which is , is greater than that of , which is . However, no information is given about . If Statement alone is assumed, it similarly follows that has greater area than , but nothing is known about the area of . If both Statements are given, however, then, by transitivity, has the greatest area of the three.

Neither statement alone is enough to determine which triangle has the greater area, as each statement gives information about only one point.

Assume both statements to be true. Since is the segment that connects the endpoints of two sides of , it is a midsegment of the triangle, whose length is half the length of the side of to which it is parallel. Therefore, the sidelength of is half that of ; it follows that is the triangle with the greater area.

The area formula for an equilateral triangle is ; that of a circle is . Both will be used here.

Assume Statement 1 alone. If we let be the radius of the circle, then, since the points on the circle include the midpoint of , the distance from center to that midpoint is , and the length of is . The area of the circle is , and that of the triangle is

.

Since , the circle has the greater area.

Assume Statement 2 alone. For simplicity's sake, assume that the triangle has sidelength 1. Then its area is . Since we only know that is the center of the circle and is outside it, it follows that the radius must be less than 1. This means that the area of the circle must be less than

Since the area falls in the range , and , we cannot tell whether the circle or the triangle has the greater area.

Since the area of an equilateral triangle is wholly dependent on its common sidelength, comparison of the lengths of the sides is all that is necessary to determine which triangle, if either, has the greater area.

If we let and be the common sidelengths of and , respectively, Statements 1 and 2, respectively, can be rewritten as:

Statement 1:

Statement 2: .

The question is now whether , , or .

Statement 1 alone is insufficient to determine which sidelength is greater; both and are easily seen to be solutions, with in the first case, and in the second. Consequently, it is possible for either triangle to have the greater sidelength and the greater area.

Statement 2 alone, however, is sufficient. If , if follows that

and

This means that . has the greater sidelength and, consequently, the greater area.

Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area.

Statement 1 alone gives that the length of one side of is greater than that on one side of .It follows that has the greater area.

Statement 2 alone gives that , from which it follows that

and

.

Again, this shows that has the greater sidelength and the greater area.

Let and be the sidelengths of and , respectively; the statements can be rewritten as:

Statement 1:

Statement 2:

Since the area of an equilateral triangle is solely dependent on the length of one side, it follows that the triangle with the greater sidelength has the greater area. The question can therefore be reduced to asking which of and , if either, is greater.

Assume Statement 1 alone. Then

,

and has the greater sidelength. It follows that its area is the greater as well.

Assume Statement 2 alone. Then

or

.

However, this is not enough to prove which triangle, if either, has the greater sidelength; if , for example, or would make this inequality true. Therefore, it is not clear whether or , if either, is greater, and it is not clear which triangle has the greater sidelength - and area.

If we let and be the sidelengths of and , respectively; the statements can be rewritten as:

Statement 1:

Statement 2:

Since the area of an equilateral triangle is dependent solely upon the length of each side, comparison of the lengths of the sides is all that is necessary to determine which triangle, has the greater area. The question can therefore be reduced to asking which of and , if either, is greater.

Statement 1 alone is not sufficient to yield an answer, as we see by examining these two scenarios:

Case 1:

Case 2:

Both cases satisfy Statement 1, but in the first case, , meaning that has greater sidelength and area than , and in the second case, , meaning the reverse. By a similar argument, Statement 2 is insufficient.

Now assume both statements to be true. The two equations together comprise a system of equations:

Multiply the second equation by and add to the first:

Now substitute back:

The sidelengths are the same, and, consequently, so are the areas of the triangles.

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.