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PSAT Math : Plane Geometry

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Example Questions

Example Question #13 : How To Find The Area Of A Rectangle

Rectangle

Note: Figure NOT drawn to scale

AB = 20, BC = 8, AD = 10, DF = 4

Give the ratio of the perimeter of Rectangle ACGF to that of Rectangle ABED .

Possible Answers:

5:3

2:1

5:2

4:3

7:5

Correct answer:

7:5

Explanation:

The perimeter of Rectangle ABED is

P = AB +BE + ED + AD

Opposite sides of a rectangle are congruent, so

AB= ED = 20, AD = BE = 10

and

P = 20 + 10 + 20 + 10 = 60

The perimeter of Rectangle ACGF is

P = AC + CG + GF + AF

Opposite sides of a rectangle are congruent, so

GF = AC = AB + BC = 20 + 8 = 28 ,

CG = AF = AD + DF = 10 + 4 = 14 ,

and

P = 28 + 14+ 28 + 14= 84

The ratio of the perimeters is

$\frac{84}{60} = \frac{84 \div 12}{60 \div 12} = \frac{7}{5}$ - that is, 7 to 5.

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Example Question #331 : Plane Geometry

Garden

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) eight feet wide throughout. What is the area of that dirt path?

Possible Answers:

$2,304\textrm{ ft}^{2}$

$1,280\textrm{ ft}^{2}$

$2,560\textrm{ ft}^{2}$

The correct area is not given among the other responses.

$1,216\textrm{ ft}^{2}$

Correct answer:

$2,304\textrm{ ft}^{2}$

Explanation:

The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is

$A = 100 \cdot 60 = 6,000$ square feet.

The inner rectangle has length and width $(100 - 2\times 8) = 84$ feet and $(60 - 2\times 8) = 44$ feet, respectively, so its area is

$A = 84 \times 44= 3,696$ square feet.

The area of the path is the difference of the two:

6,000 - 3,696= 2,304 square feet.

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Example Question #15 : How To Find The Area Of A Rectangle

Garden

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange). The dirt path is seven feet wide throughout. Which of the following polynomials gives the area of the dirt path in square feet?

Possible Answers:

42x

42x - 196

21x-49

42x - 98

21x

Correct answer:

42x - 196

Explanation:

The area of the dirt path is the difference between the areas of the outer and inner rectangles.

The outer rectangle has area

$A = 2x \cdot x = 2x^{2}$

The area of the inner rectangle can be found as follows:

The length of the garden is $2 \times 7= 14$ feet less than that of the entire lot, or

L = 2x-14 ;

The width of the garden is $2 \times 7= 14$ less than that of the entire lot, or

W = x-14 ;

The area of the garden is their product:

A = LW = (2x-14)(x - 14)

$=2x \cdot x - 2x \cdot 14 - 14 \cdot x + 14 \cdot 14$

$= 2x ^{2} - 28x - 14 x + 196$

$= 2x ^{2} - 42 x + 196$

Now, subtract the areas:

$2x ^{2} - \left (2x ^{2} - 42 x + 196 \right ) = 2x ^{2} - 2x ^{2} + 42 x - 196 = 42x - 196$

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Example Question #41 : Rectangles

Two circles of a radius of {2}'' each sit inside a square with a side length of {8}'' . If the circles do not overlap, what is the area outside of the circles, but within the square?

Possible Answers:

$16+8\pi$

$36-6\pi$

$64-8\pi$

$36-8\pi$

$36-4\pi$

Correct answer:

$64-8\pi$

Explanation:

The area of a square = $\dpi{100} \small side^{2}$

The area of a circle is $\dpi{100} \small \pi r^{2}$

Area = Area of Square $\dpi{100} \small -$ 2(Area of Circle) =

$8^{2}-2\cdot \pi (2)^{2}$

$64-8\pi$

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Example Question #331 : Plane Geometry

If the area Rectangle A is $25\%$ larger than Rectangle B and the sides of Rectangle A are and , what is the area of Rectangle B?

Possible Answers:

Correct answer:

Explanation:

$A = \text{length}\: \times \:\text{width} = 10 \times 6 = 60$

A = 1.25 B

60 = 1.25B

$B =\frac{60}{1.25} = 48$

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Example Question #1 : Parallelograms

ABCD is a parallelogram. BD = 5. The angles of triangle ABD are all equal. What is the perimeter of the parallelogram?

Figure_1

Possible Answers:

$10\sqrt{3}$

$15\sqrt{3}$

$11\sqrt{2}$

Correct answer:

Explanation:

If all of the angles in triangle ABD are equal and line BD divides the parallelogram, then all angles in triangle BDC must be equal as well.

We now have two equilateral triangles, so all sides of the triangles will be equal.

All sides therefore equal 5.

5+5+5+5 = 20

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PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #13 : How To Find The Area Of A Rectangle

Example Question #331 : Plane Geometry

Example Question #15 : How To Find The Area Of A Rectangle

Example Question #41 : Rectangles

Example Question #331 : Plane Geometry

Example Question #1 : Parallelograms

All PSAT Math Resources

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