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SAT II Math I : Geometry

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

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Example Question #1 : Geometry

Swimming_pool

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:

$315 \textrm{ yd}^{2}$

$216 \textrm{ yd}^{2}$

$252 \textrm{ yd}^{2}$

$198\textrm{ yd}^{2}$

$270 \textrm{ yd}^{2}$

Correct answer:

$270 \textrm{ yd}^{2}$

Explanation:

Three feet make a yard, so the length and width of the pool are $\frac{50}{3} = 16 \frac{2}{3}$ yards and $\frac{35}{3} = 11 \frac{2}{3}$ 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 $17 \frac{2}{3}$ yards by $12 \frac{2}{3}$ 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

$18 \times 15 = 270$ square yards.

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Example Question #2 : Area

Triangle

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:

$33 \frac{1}{3} \%$

$25 \%$

Insufficient information is given to answer the question.

$50 \%$

$44\frac{4}{9} \%$

Correct answer:

$44\frac{4}{9} \%$

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 3:2 . The ratio of the areas is the square of this, or $3^{2}:2^{2}$ , or 9:4 .

The blue triangle is therefore $\frac{4}{9}$ of the entire triangle, or $\frac{4}{9} \times 100 = 44 \frac{4}{9} \%$ of it.

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Example Question #1 : 2 Dimensional Geometry

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the area of $\Delta ABC$ ?

Possible Answers:

210

144

Insufficient information is given to answer the problem.

225

180

Correct answer:

180

Explanation:

If we see hypotenuse $\overline{AC}$ as the base of the large triangle, then we can look at the segment perpendicular to it, $\overline{BD}$ , as its altitude. Therefore, the area of $\Delta ABC$ is

$A = \frac{1}{2} (AC) (BD)$

AC = AD + DC = 6 + 24 = 30 .

, 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:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$

The area of $\Delta ABC$ is therefore

$A = \frac{1}{2} (AC) (BD) = \frac{1}{2} \cdot 30 \cdot 12 = 180$

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Example Question #1 : 2 Dimensional Geometry

Swimming_pool

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:

$160\textrm{ yd}^{2}$

$180\textrm{ yd}^{2}$

$200 \textrm{ yd}^{2}$

$190\textrm{ yd}^{2}$

$170\textrm{ yd}^{2}$

Correct answer:

$200 \textrm{ yd}^{2}$

Explanation:

Three feet make a yard, so the length and width of the pool are $\frac{50}{3}$ yards and $\frac{35}{3}$ yards; the area of the pool, and that of the tarp needed to cover it, must be the product of these dimensions, or

$\frac{50}{3} \times \frac{35}{3}= \frac{1,750}{9} = 194 \frac{4}{9}$ 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.

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Example Question #5 : Area

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. In terms of area, $\Delta ADB$ is what fraction of $\Delta ABC$ ?

Possible Answers:

$\frac{1}{6}$

Insufficient information is given to answer this question.

$\frac{1}{5}$

$\frac{1}{4}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{5}$

Explanation:

The area of $\Delta ADB$ , being right, is half the products of its legs, which is:

$\frac{1}{2} (AD) (BD) = \frac{1}{2} \times 6 (BD) = 3 (BD)$

The area of $\Delta ABC$ is one half the product of its base and height; we can use its hypotenuse AC = AD + DC = 6 + 24 = 30 as the base and as the height, so this area is

$\frac{1}{2} (AC) (BD) = \frac{1}{2} \times 30 (BD) = 15 (BD)$

Therefore, in terms of area, $\Delta ADB$ is $\frac{3 (BD)}{15 (BD)} = \frac{3}{15} = \frac{1}{5}$ of $\Delta ABC$ .

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Example Question #1 : Geometry

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of $\Delta BDC$ to that of $\Delta ADB$ .

Possible Answers:

5:2

2:1

Insufficient information is given to answer the question.

4 :1

3:1

Correct answer:

4 :1

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:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{6 \cdot 24} = \sqrt{144} = 12$ .

The areas of $\Delta BDC$ and $\Delta ADB$ , each being right, are half the products of their legs, so:

The area of $\Delta ADB$ is $\frac{1}{2} \times (AD) (BD) = \frac{1}{2} \times 6 \times (BD)= 3(BD)$

The area of $\Delta BDC$ is $\frac{1}{2} \times (CD) (BD) = \frac{1}{2} \times 24 \times (BD)= 12(BD)$

The ratio of the areas is $\frac{12 (BD)}{3 (BD)} = \frac{12 }{3 } =\frac{4}{1}$ - that is, 4 to 1.

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Example Question #1 : Geometry

Thingy

Note: figure NOT drawn to scale.

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

Possible Answers:

Insufficient information is given to calculate the area.

149

109

139

189

Correct answer:

109

Explanation:

Polygon ABCDE is a composite of $\Delta ABE$ and Square BCDE ; its area is the sum of the areas of the two figures.

$\Delta ABE$ is a right triangle; its area is half the product of its legs, which is

$A = \frac{1}{2} \cdot 5 \cdot 8 = 20$

$\overline{BE }$ is both one side of Square BCDE and the hypotenuse of $\Delta ABE$ ; its hypotenuse can be calculated from the lengths of the legs using the Pythagorean Theorem:

$BE = \sqrt{5^{2}+8^{2}} = \sqrt{25 + 64 } = \sqrt{89}$ .

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

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

20 + 89 = 109 .

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Example Question #1 : Area

Find the area of a circle with a diameter of .

Possible Answers:

$32\pi$

$64\pi$

$81\pi$

$16\pi$

$256\pi$

Correct answer:

$64\pi$

Explanation:

Write the formula for the area of a circle.

$A_{circle}= \pi r^2 = \frac{\pi d^2}{4}$

Substitute the diameter and solve.

$\frac{\pi (16)^2}{4} = \frac{\pi (16)(16)}{4}=\pi (16)(4)= 64\pi$

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Example Question #9 : Area

Rectangle example

Figure not drawn to scale.

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

Possible Answers:

84 in²

144 in²

36 in²

70 in²

72 in²

Correct answer:

72 in²

Explanation:

Rectangle example

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:

$\small Perimeter= 2(length)+2(width)$

$\small 36=2(12)+2w$

$\small 36=24+2w$

$\small 12=2w$

$\small w=6$

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.

$\small Area=(length)(width)$

$\small (12)(6)=72$

The area of the rectangle is 72 in²

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Example Question #1 : Area

Which of the following shapes is a kite?

Shapes

Possible Answers:

A, B

Correct answer:

A, B

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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SAT II Math I : Geometry

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Geometry

Example Question #2 : Area

Example Question #1 : 2 Dimensional Geometry

Example Question #1 : 2 Dimensional Geometry

Example Question #5 : Area

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #1 : Area

Example Question #9 : Area

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

Side w is 6 in long.

The area of the rectangle is 72 in²

Example Question #1 : Area

All SAT II Math I Resources

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HIGH SCHOOL

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SAT II Math I : Geometry

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Geometry

Example Question #2 : Area

Example Question #1 : 2 Dimensional Geometry

Example Question #1 : 2 Dimensional Geometry

Example Question #5 : Area

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #1 : Area

Example Question #9 : Area

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

Side w is 6 in long.

The area of the rectangle is 72 in2

Example Question #1 : Area

All SAT II Math I Resources

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The area of the rectangle is 72 in²