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## Example Questions

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

What is the area of ?

(1) The height is 5.

(2) The base is 4.

**Possible Answers:**

Statement 1 alone is sufficient

Statement 2 alone is sufficient

Statements 1 and 2 together are not sufficient

Both statements taken together are sufficient

Each statement alone is sufficient

**Correct answer:**

Both statements taken together are sufficient

To find an area of a triangle we need the length of the height and the length of the corresponding basis.

Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.

Hence, both statements together are sufficient.

### Example Question #2 : Dsq: Calculating The Area Of An Equilateral Triangle

is an equilateral triangle inscribed in the circle. What is the area of triangle ?

(1) The area of the circle is .

(2) The area of the circle minus the area of the triangle is .

**Possible Answers:**

Statement 1 alone is sufficient.

Each statement alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 taken together are sufficient.

Statement 2 alone is sufficient.

**Correct answer:**

Statement 1 alone is sufficient.

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.

### Example Question #3 : Dsq: Calculating The Area Of An Equilateral Triangle

Consider .

I) .

II) Side light years long.

What is the area of ?

**Possible Answers:**

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.

Neither statement is sufficient to answer the question. More information is needed.

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

**Correct answer:**

Both statements are needed to answer the question.

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.

### Example Question #4 : Dsq: Calculating The Area Of An Equilateral Triangle

Given an equilateral triangle and Square , which, if either, has the greater area?

Statement 1: and are located on the same circle.

Statement 2: Each diagonal of Square has length times the height of .

**Possible Answers:**

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.

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.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

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.

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

Which, if either, of equilateral triangles and , has the greater area?

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.

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.

**Correct answer:**

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

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.

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

Which figure, if either, has the greater area: equilateral triangle or a given circle with center ?

Statement 1: The midpoint of is on the circle.

Statement 2: is outside the circle.

**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.

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.

**Correct answer:**

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.

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

Given three equilateral triangles , , and , which has the greatest area?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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.

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

Given two equilateral triangles and , which, if either, has the greater area?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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.

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

Given two equilateral triangles and , which has the greater area?

Statement 1: is the midpoint of .

Statement 2: is the midpoint of .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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.

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

Which, if either, of equilateral triangles and , has the greater area?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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.

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