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PSAT Math : Plane Geometry

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Example Questions

Example Question #5 : Rectangles

Rectangles

Note: Figure NOT drawn to scale.

In the above figure,

$\textup{Rect } ABCD \sim \textup{Rect } DBEF$ .

AB = 12, AD = 6 .

Give the area of Polygon ABEFD .

Possible Answers:

162

126

$36 + 18\sqrt{5}$

$72+ 18\sqrt{5}$

Correct answer:

126

Explanation:

Polygon ABEFD can be seen as a composite of right $\Delta ABD$ and $\textup{Rect } DBEF$ , so we calculate the individual areas and add them.

The area of $\Delta ABD$ is half the product of legs and A D :

$\frac{1}{2} \cdot AB \cdot AD = \frac{1}{2} \cdot 12 \cdot 6 = 36$

Now we find the area of $\textup{Rect } DBEF$ . We can do this by first finding using the Pythagorean Theorem:

$DB = \sqrt{(AD)^{2}+(AB)^{2}}$

$DB = \sqrt{6^{2}+12^{2}} = \sqrt{36+144} = \sqrt{180} = \sqrt{36} \cdot \sqrt{5} = 6 \sqrt{5}$

The similarity of $\textup{Rect } DBEF$ to $\textup{Rect } ABCD$ implies

$\frac{DF}{AD}= \frac{DB}{AB}$

$\frac{DF}{6}= \frac{6\sqrt{5}}{12}$

$DF= \frac{6 \cdot 6\sqrt{5}}{12} = \frac{3 6\sqrt{5}}{12} = 3\sqrt{5}$

The area of $\textup{Rect } DBEF$ is the product of and :

$DB \cdot DF = 6 \sqrt{5} \cdot 3 \sqrt{5}= 6\cdot 3 \cdot \sqrt{5} \cdot \sqrt{5}= 18 \cdot 5 = 90$

Now add: 90 + 36 = 126 , the correct response.

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Example Question #1 : How To Find If Rectangles Are Similar

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above figure.

$\textup{Rect }ABGF \sim \textup{Rect }ACDE$

AB = 7 and BC = 3 .

What percent of $\textup{Rect }ACDE$ has been shaded brown ?

Possible Answers:

Insufficient information is given to answer the problem.

$51 \%$

$30 \%$

$49 \%$

$57 \frac{1}{7} \%$

Correct answer:

$51 \%$

Explanation:

AB = 7 and AC = AB + BC = 7 + 3 = 10 , so the similarity ratio of $\textup{Rect }ACDE$ to $\textup{Rect }ABGF$ is 10 to 7. The ratio of the areas is the square of this, or

$10^{2}:7^{2}$

100:49

Therefore, $\textup{Rect }ABGF$ comprises $\frac{49}{100} = 49 \%$ of $\textup{Rect }ABGF$ , and the remainder of the rectangle - the brown region - is 51% of $\textup{Rect }ABGF$ .

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Example Question #311 : Plane Geometry

Rectangles

Note: figure NOT drawn to scale.

Refer to the above figure.

$\textup{Rect }ABGF \sim \textup{Rect }ACDE$

AB = 12 , BC = 3 , AE = 10 .

Give the area of $\textup{Rect }ABGF$ .

Possible Answers:

108

216

120

Correct answer:

Explanation:

$\textup{Rect }ABGF \sim \textup{Rect }ACDE$ , so the sides are in proportion - that is,

$\frac{AF}{AB}= \frac{AE}{AC}$

AC = AB + BC = 12 + 3 = 15

Set

AB = 12 , AC = 15 , AE = 10 and solve for :

$\frac{AF}{12}= \frac{10}{15}$

$AF = \frac{10}{15} \cdot 12 = 8$

$\textup{Rect }ABGF$ has area

$AB \cdot AF = 12 \cdot 8 = 96$

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Example Question #1 : How To Find The Perimeter Of A Rectangle

A rectangle has a width of 2x. If the length is five more than 150% of the width, what is the perimeter of the rectangle?

Possible Answers:

5x + 10

6x² + 5

10(x + 1)

5x + 5

6x² + 10x

Correct answer:

10(x + 1)

Explanation:

Given that w = 2x and l = 1.5w + 5, a substitution will show that l = 1.5(2x) + 5 = 3x + 5.

P = 2w + 2l = 2(2x) + 2(3x + 5) = 4x + 6x + 10 = 10x + 10 = 10(x + 1)

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Example Question #11 : Rectangles

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). The dirt path is seven feet wide throughout. Which of the following polynomials gives the perimeter of the garden in feet?

Possible Answers:

6x - 28

6x - 56

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

$2x^{2}-21x + 49$

6x-14

Correct answer:

6x - 56

Explanation:

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

L = 2x-14 ;

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

W = x-14 .

The perimeter, in feet, is twice the sum of the two:

P = 2 (L + W)

= 2 (2x-14 + x - 14 )= 2 (3x-28) = 6x - 56

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Example Question #11 : Rectangles

Garden

Note: Figure NOT drawn to scale

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

Possible Answers:

$284\textup{ ft}$

$118 \textup{ ft}$

$134 \textup{ ft}$

$268 \textup{ ft}$

$236 \textup{ ft}$

Correct answer:

$236 \textup{ ft}$

Explanation:

The inner square, which represents the garden, has sidelength $(75 - 2\times 8) = 59$ feet, so its perimeter is four times this:

$P = 4 \times 59 = 236$ feet.

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Example Question #31 : Quadrilaterals

Farmer Dave has a rectangular field that is 50 yards wide and 40 yards long. He wants to enclose the field with a wire fence. How much wire does Farmer Dave need?

Possible Answers:

180 yards

170 yards

160 yards

200 yards

210 yards

Correct answer:

180 yards

Explanation:

To solve this problem, find the perimeter of the rectangle. There are two sides that each measure 50 yards and two sides that each measure 40 yards. Together these four sides measure 180 yards.

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Example Question #1 : How To Find The Perimeter Of A Rectangle

A rectangular garden has an area of $80\: \mbox{m^2}$ . Its length is meters longer than its width. How much fencing is needed to enclose the garden?

Possible Answers:

$54\: \mbox{meters}$

$24\: \mbox{meters}$

$40\: \mbox{meters}$

$36\: \mbox{meters}$

$18\: \mbox{meters}$

Correct answer:

$36\: \mbox{meters}$

Explanation:

We define the variables as $w = \mbox{width}$ and $l = \mbox{length} = w + 2$ .

We substitute these values into the equation for the area of a rectangle and get $A_{\mbox{rectangle} }= wl = w(w + 2) = 80$ .

w^2 +2w-80=0

(w - 8)(w + 10) = 0

w = 8 or w = -10

Lengths cannot be negative, so the only correct answer is w = 8 . If w = 8 , then l = 10 .

Therefore, $\mbox{perimeter }= 2w + 2l = 16 + 20 = 36\:\mbox{m}$ .

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

A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?

Possible Answers:

Correct answer:

Explanation:

We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)

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Example Question #1 : Rectangles

The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?

Possible Answers:

$4800

$1080

$5400

$1920

$2700

Correct answer:

$2700

Explanation:

The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.

Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.

10 feet = 10 x 12 inches = 120 inches

12 feet = 12 x 12 inches = 144 inches

This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in² = 17280 in².

Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in².

Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in², or 16 in².

If there are 17280 in² to be covered, and each tile is 16 in², then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.

The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.

The total cost = 1080 x 2.50 dollars = 2700 dollars.

The answer is $2700.

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PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #5 : Rectangles

Example Question #1 : How To Find If Rectangles Are Similar

Example Question #311 : Plane Geometry

Example Question #1 : How To Find The Perimeter Of A Rectangle

Example Question #11 : Rectangles

Example Question #11 : Rectangles

Example Question #31 : Quadrilaterals

Example Question #1 : How To Find The Perimeter Of A Rectangle

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

Example Question #1 : Rectangles

All PSAT Math Resources

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