# SAT II Math I : 2-Dimensional Geometry

## Example Questions

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### Example Question #1 : 2 Dimensional Geometry

The above figure depicts the rectangular swimming pool at an apartment. The apartment manager needs to purchase a tarp that will cover this pool completely. However, because of the cutting device the pool store uses, the length and the width of the tarp must each be a multiple of three yards. Also, the tarp must be at least one yard longer and one yard wider than the pool.

What will be the minimum area of the tarp the manager purchases?

Possible Answers:

Correct answer:

Explanation:

Three feet make a yard, so the length and width of the pool are  yards and  yards, respectively. Since the dimensions of the tarp must exceed those of the pool by at least one yard, the tarp must be at least  yards by  yards; but since both dimensions must be multiples of three yards, we take the next multiple of three for each.

The tarp must be 18 yards by 15 yards, so the area of the tarp is the product of these dimensions, or

square yards.

### Example Question #1 : Geometry

Note: Figure NOT drawn to scale.

Refer to the figure above, which shows a square inscribed inside a large triangle. What percent of the entire triangle has been shaded blue?

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

The shaded portion of the entire triangle is similar to the entire large triangle by the Angle-Angle postulate, so sides are in proportion. The short leg of the blue triangle has length 20; that of the large triangle, 30. Therefore, the similarity ratio is . The ratio of the areas is the square of this, or , or

The blue triangle is therefore  of the entire triangle, or  of it.

### Example Question #3 : 2 Dimensional Geometry

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the area of ?

Possible Answers:

Insufficient information is given to answer the problem.

Correct answer:

Explanation:

If we see hypotenuse  as the base of the large triangle, then we can look at the segment perpendicular to it, , as its altitude. Therefore, the area of  is

.

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, the square root of the product of the two:

The area of  is therefore

### Example Question #4 : 2 Dimensional Geometry

The above figure depicts the rectangular swimming pool at an apartment. The apartment manager needs to purchase a tarp that will cover this pool completely, but the store will only sell the material in multiples of ten square yards. How many square yards will the manager need to buy?

Possible Answers:

Correct answer:

Explanation:

Three feet make a yard, so the length and width of the pool are  yards and  yards; the area of the pool, and that of the tarp needed to cover it, must be the product of these dimensions, or

square yards.

The manager will need to buy a number of square yards of tarp equal to the next highest multiple of ten, which is 200 square yards.

### Example Question #1 : Geometry

Note: Figure NOT drawn to scale.

Refer to the above diagram. In terms of area,  is what fraction of ?

Possible Answers:

Insufficient information is given to answer this question.

Correct answer:

Explanation:

The area of , being right, is half the products of its legs, which is:

The area of  is one half the product of its base and height; we can use its hypotenuse  as the base and  as the height, so this area is

Therefore, in terms of area,  is  of .

### Example Question #6 : 2 Dimensional Geometry

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of  to that of .

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

.

The areas of  and , each being right, are half the products of their legs, so:

The area of  is

The area of  is

The ratio of the areas is  - that is, 4 to 1.

### Example Question #7 : 2 Dimensional Geometry

Note: figure NOT drawn to scale.

Refer to the above figure. Quadrilateral  is a square. What is the area of Polygon ?

Possible Answers:

Insufficient information is given to calculate the area.

Correct answer:

Explanation:

Polygon  is a composite of  and Square ; its area is the sum of the areas of the two figures.

is a right triangle; its area is half the product of its legs, which is

is both one side of Square  and the hypotenuse of ;  its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

.

Square  has area the square of this, which is 89.

Polygon  has as its area the sum of these two areas:

.

### Example Question #8 : 2 Dimensional Geometry

Find the area of a circle with a diameter of .

Possible Answers:

Correct answer:

Explanation:

Write the formula for the area of a circle.

Substitute the diameter and solve.

### Example Question #9 : 2 Dimensional Geometry

Figure not drawn to scale.

# Find the area of the rectangle above when the perimeter is 36 in.

Possible Answers:

72 in2

70 in2

84 in2

144 in2

36 in2

Correct answer:

72 in2

Explanation:

Because we know the perimeter is 36 inches, we can determine the length of side w based on the equation of the perimeter of a rectangle:

### Side w is 6 in long.

Now that we know that side w is 6 inches long, we have everythinng we need to calculate the area of the rectangle.

### Example Question #1 : How To Find The Area Of A Kite

Which of the following shapes is a kite?

Possible Answers:

Correct answer:

Explanation:

A kite is a four-sided shape with straight sides that has two pairs of sides. Each pair of adjacent sides are equal in length. A square is also considered a kite.

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