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## Example Questions

### Example Question #9 : Sectors

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

**Possible Answers:**

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.

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

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is ; the total area of the two black quarter circles is , so the area of the white region is their difference,

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is centimeters, then we can deduce using this equation:

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

### Example Question #1 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The circle has circumference .

Statement 2:

**Possible Answers:**

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.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by to yield the radius, and that can be substituted for in the formula to find the area. However, it provides no clue that might yield .

Statement 2 alone asserts that . This is an inscribed angle that intercepts the arc ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is and the area is . The shaded sector is of the circle, so the area can be calculated to be .

### Example Question #1 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2:

**Possible Answers:**

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.

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.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Statement 1 alone asserts that . This is an inscribed angle that intercepts the arc ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or .

Statement 2 alone asserts that . By angle addition, .

Either statement alone tells us that the shaded sector is of the circle, and that the white sector is of it; it can be subsequently calculated that the ratio of the areas is , or .

### Example Question #3 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

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:**

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that . is an inscribed angle that intercepts the arc ; therefore, the arc - and the central angle that intercepts it - has twice its measure, or . From angle addition, this can be subtracted from to yield the measure of central angle of the shaded sector, which is . That makes that sector of the circle. The white sector is of the circle, and the ratio of the areas can be determined to be , or .

### Example Question #2 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: .

Statement 2: The circle has circumference .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by to yield radius , and that can be substituted for in the formula to find the area: .

However, it provides no clue that might yield .

From Statement 2 alone, we can find . , an inscribed angle, intercepts an arc twice its measure - this arc is , which has measure . , the corresponding minor arc, will have measure . This gives us , but no clue that yields the area.

Now assume both statements are true. The area is and the shaded sector is of the circle, so the area can be calculated to be .

### Example Question #6 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle has area .

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Assume Statement 1 alone. No clues are given about the measure of , so that of , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle , or, subsequently, , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and be the measure of . Then:

and

The statements can be simplified as

and

From these two statements:

; the second statement can be solved for :

.

, so .

Since , the circle has area . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

.

### Example Question #5 : Dsq: Calculating The Area Of A Sector

What is the area of a sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is inches.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The area of a sector of radius is

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the arc is

Given that length, you can find the radius:

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

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