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PSAT Math : Plane Geometry

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Example Question #6 : Quadrilaterals

Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?

Possible Answers:

Correct answer:

Explanation:

Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.

Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2l + 2w.

We also know that if s is the length of a side of a square, then the perimeter is 4s, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.

2l + 2w = 4s

If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:

Recsquare1

Recsquare2

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

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

Possible Answers:

6x² + 10x

5x + 10

5x + 5

10(x + 1)

6x² + 5

Correct answer:

6x² + 10x

Explanation:

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

A = lw = (3x + 5)(2x) = 6x² + 10x

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Example Question #181 : Geometry

Para-rec1

Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – x² , and points C and D lie on the graph of y = x² – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?

Possible Answers:

352

176

272

544

Correct answer:

176

Explanation:

Para-rec2

Para-rec3

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Example Question #21 : Rectangles

George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 $ft^{2}$ and costs $40. One quart of paint covers 100 $ft^{2}$ and costs $15. How much money will he spend on the blue paint?

Possible Answers:

$\$40$

$\$30$

$\$70$

$\$80$

$\$55$

Correct answer:

$\$70$

Explanation:

The area of the walls is given by $A = 2wh + 2lh = 2(12)(10) + 2(15)(10) = 540 ft^{2}$

One gallon of paint covers 400 $ft^{2}$ and the remaining 140 $ft^{2}$ would be covered by two quarts.

So one gallon and two quarts of paint would cost $\$40 + \$15 + \$15 = \$70$

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Example Question #4 : How To Find The Area Of A Rectangle

Daisy gets new carpet for her rectangluar room. Her floor is $21\ ft \times 24\ ft$ . The carpet sells for $5 per square yard. How much did she spend on her carpet?

Possible Answers:

$\$225$

$\$350$

$\$120$

$\$310$

$\$280$

Correct answer:

$\$280$

Explanation:

Since $3\ ft=1\ yd$ the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost $5\cdot 56=\$280$ .

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

The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?

Possible Answers:

$150\ ft^{2}$

$125\ ft^{2}$

$75\ ft^{2}$

$100\ ft^{2}$

$50\ ft^{2}$

Correct answer:

$75\ ft^{2}$

Explanation:

For a rectangle, $P=2w+2l$ and $A=lw$ where $w$ is the width and $l$ is the length.

Let $x=width$ and $2x+5=length$ .

So the equation to solve becomes $40=2x+2(2x+5)$ or $40=6x+10$ .

Thus $x=5\ ft$ and $2x+5=15\ ft$ , so the area is $75\ ft^{2}$ .

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Example Question #1 : Rectangles

The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?

Possible Answers:

768 ft²

1536 ft²

192 ft²

1536 ft²

2464 ft²

Correct answer:

1536 ft²

Explanation:

Glass Area per Window = 8 ft x 6 ft = 48 ft²

Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows

Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft²

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Example Question #21 : Rectangles

Rectangle

Note: Figure NOT drawn to scale

AB = 13, BC = 7, AD = 9, DF = 3

What percent of Rectangle ACGF is pink?

Possible Answers:

$48\frac{3}{4} \%$

$50 \%$

$46\frac{2}{3} \%$

$55 \frac{5}{9} \%$

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

Correct answer:

$48\frac{3}{4} \%$

Explanation:

The pink region is Rectangle ABED . Its length and width are

L = AB = 13

W = AD = 9

so its area is the product of these, or

$A = 13 \times 9 = 117$ .

The length and width of Rectangle ACGF are

L = AC = AB + BC = 13+7 = 20

W =AF = AD + DF=9+3 = 12

so its area is the product of these, or

$20 \times 12 = 240$ .

We want to know what percent 117 is of 240, which can be answered as follows:

$\frac{117}{240} \times 100 \% = 48\frac{3}{4} \%$

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Example Question #322 : Plane Geometry

Rectangle

Note: Figure NOT drawn to scale

Refer to the above diagram.

40% of Rectangle ACGF is pink. AB = 20, AD = 15, BC = 10 .

Evaluate .

Possible Answers:

DF = 10

DF = 5

DF = 8

DF = 12

DF = 15

Correct answer:

DF = 10

Explanation:

Rectangle ABED has length L = AB = 20 and width W = AD = 15 , so it has area

$LW = 20 \times 15 = 300$ .

300 is 40% of, or 0.40 times, the area of Rectangle ACGF , which we will call . We can determine as follows:

0.40 A = 300

$A = 300 \div 0.4 = 750$ .

The length of Rectangle ACGF is

L = AC = AB + BC = 20 + 10 = 30 ,

so its width is

$W = AF = 750 \div 30 = 25$ .

Since

W = AF= AD + DF ,

25 = 15 + DF

DF = 10

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Example Question #321 : Plane Geometry

Rectangle

Note: Figure NOT drawn to scale

AB = 24, BC = 6, AD = 12, DF = 3

What percent of Rectangle ACGF is white?

Possible Answers:

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

$40 \%$

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

$36 \%$

$25 \%$

Correct answer:

$36 \%$

Explanation:

The pink region is Rectangle ABED . Its length and width are

L = AB = 24

W = AD = 12

so its area is the product of these, or

$24 \times 12 = 288$ .

The length and width of Rectangle ACGF are

L = AC = AB + BC = 24 + 6 =30

W =AF = AD + DF= 12 + 3 = 15

so its area is the product of these, or

$30 \times 15 = 450$ .

The white region is Rectangle ABED cut from Rectangle ACGF , so its area is the difference of the two:

450 - 288 = 162 .

So we want to know what percent 162 is of 450, which can be answered as follows:

$\frac{162}{450} \times 100 \% = 36 \%$

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PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #6 : Quadrilaterals

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

Example Question #181 : Geometry

Example Question #21 : Rectangles

Example Question #4 : How To Find The Area Of A Rectangle

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

Example Question #1 : Rectangles

Example Question #21 : Rectangles

Example Question #322 : Plane Geometry

Example Question #321 : Plane Geometry

All PSAT Math Resources

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