SSAT Middle Level Math : Rectangles

Study concepts, example questions & explanations for SSAT Middle Level Math

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Example Questions

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

The width of a rectangle is half of its length.  If the width is given as \displaystyle wwhat is the perimeter of the rectangle in terms of \displaystyle w?

Possible Answers:

\displaystyle 4w+6

\displaystyle 6w+2

\displaystyle 4w

\displaystyle 2w+4

\displaystyle 6w

Correct answer:

\displaystyle 6w

Explanation:

The sum of the widths is \displaystyle 2wand since the width is half the length, each length is \displaystyle 2wSince there are 2 lengths we get a total perimeter of \displaystyle 6w.

Example Question #2 : How To Find The Perimeter Of A Rectangle

L_lawn

The above figure shows the size and shape of a yard that is to be surrounded by some fence. How many feet of fence will be needed?

Note: all sides meet at right angles.

Possible Answers:

\displaystyle 650 \textrm{ ft}

\displaystyle 565 \textrm{ ft}

\displaystyle 665 \textrm{ ft}

\displaystyle 750 \textrm{ ft}

\displaystyle 935 \textrm{ ft}

Correct answer:

\displaystyle 750 \textrm{ ft}

Explanation:

The best way to see that 750 feet of fence are needed is to look at this augmented diagram.

Note that two of the sides are extended to form a smaller rectangle whose sides can be deduced by subtraction. Since opposite sides of a rectangle are congruent, this allows us to fill in the two missing sidelengths of the original figure.

L_lawn_2

Now add: \displaystyle 150+110+70+115+80+225 = 750 \textrm{ft}

Example Question #3 : How To Find The Perimeter Of A Rectangle

Rectangle

 

Give the perimeter of the rectangle in the above diagram.

Possible Answers:

\displaystyle 76 \textrm{ cm}

\displaystyle 19 \textrm{ cm}

\displaystyle 38 \textrm{ cm}

\displaystyle 40.32 \textrm{ cm}

\displaystyle 80.64 \textrm{ cm}

Correct answer:

\displaystyle 38 \textrm{ cm}

Explanation:

The perimeter of a rectangle is the sum of the length and the width, multiplied by 2:

\displaystyle 2 (12.6 + 6.4) = 2 \cdot 19 = 38

The rectangle has a perimeter of 38 centimeters.

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

Rectangle

Give the perimeter of the rectangle in the above diagram.

Possible Answers:

\displaystyle 42 \textrm{ in}

\displaystyle 13 \frac{1}{10} \textrm{ in}

\displaystyle 52 \frac{2}{5} \textrm{ in}

\displaystyle 84 \textrm{ in}

\displaystyle 26 \frac{1}{5} \textrm{ in}

Correct answer:

\displaystyle 26 \frac{1}{5} \textrm{ in}

Explanation:

The perimeter of a rectangle can be calculated by multiplying two by the sum of the length and width of the rectangle.

\displaystyle 2\left ( 7 \frac{1}{2} +5 \frac{3}{5} \right )

\displaystyle =2 \left (\frac{7 \cdot 2 + 1}{2} + \frac{5\cdot 5 + 3}{5} \right )

\displaystyle = 2 \left (\frac{15}{2} + \frac{28}{5} \right )

\displaystyle = 2 \left (\frac{15 \times 5}{2 \times 5} + \frac{2 \times28}{2 \times5} \right )

\displaystyle = 2 \left (\frac{75}{10} + \frac{56}{10} \right ) = \frac{2}{1} \cdot \frac{131}{10} = \frac{131}{5} = 26 \frac{1}{5}

The perimeter of the rectangle is \displaystyle 26 \frac{1}{5} inches.

Example Question #3 : How To Find The Perimeter Of A Rectangle

Rectangle_1

Figure NOT drawn to scale.

Give the perimeter of the green polygon in the above figure.

Possible Answers:

\displaystyle 230

\displaystyle 190

\displaystyle 260

\displaystyle 380

The perimeter cannot be determined from the information given.

 

Correct answer:

\displaystyle 230

Explanation:

Since opposite sides of a rectangle have the same measure, the missing sidelengths can be calculated as in the diagram below:

Rectangle_2

The sidelengths of the green polygon can now be added to find the perimeter:

\displaystyle P= 35 +60 + 35 + 15 + 20 + 30 + 20 + 15 = 230

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

The width of a rectangle is one-third of its length. If the width is given as \displaystyle w what is the perimeter of the rectangle in terms of \displaystyle w?

Possible Answers:

\displaystyle 2w+\frac{1}{3}w

\displaystyle 6w

\displaystyle 3w

\displaystyle 8w

\displaystyle 2w

Correct answer:

\displaystyle 8w

Explanation:

The perimeter of a rectangle is the sum of its sides.

The sum of the widths is \displaystyle 2w and since the width is one-third of the length, each length is \displaystyle 3w. Since there are \displaystyle 2 lengths we get a total of \displaystyle 6w. Widths + lengths = \displaystyle 2w + 6w = 8w

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

You are given equilateral triangle \displaystyle \Delta ABC and Rectangle \displaystyle ACDE

with \displaystyle AB = 25, CD = 40.

What is the perimeter of Rectangle \displaystyle ACDE ?

Possible Answers:

\displaystyle 160

\displaystyle 180

\displaystyle 130

\displaystyle 115

Correct answer:

\displaystyle 130

Explanation:

\displaystyle \Delta ABC is equilateral, so \displaystyle AC = AB = 25.

Also, since opposite sides of a rectangle are congruent, 

\displaystyle DE = AC = 25 and \displaystyle AE = CD = 40

The perimeter of Rectangle \displaystyle ACDE is 

\displaystyle AC + CD +DE + AE = 25 + 40 + 25 + 40 = 130

Example Question #2 : How To Find The Perimeter Of The Rectangle

A hectare is a unit of area equal to 10,000 square meters.

A 150-hectare plot of land is rectangular and is 1.2 kilometers in width. Give the perimeter of this land.

Possible Answers:

\displaystyle 2.45 \textrm{ km}

\displaystyle 2.7 \textrm{ km}

\displaystyle 5.4 \textrm{ km}

\displaystyle 4.9 \textrm{ km}

Correct answer:

\displaystyle 4.9 \textrm{ km}

Explanation:

150 hectares is equal to \displaystyle 150 \times 10,000 = 1,500,000 square meters.

The width of this land is 1.2 kilometers, or \displaystyle 1.2 \times 1,000 =1,200 meters. Divide the area by the width to get:

\displaystyle 1,500,000 \div 1,200 = 1,250 meters

The perimeter of the land is 

\displaystyle 2 \left ( 1,200 + 1,250\right ) = 4,900 meters, or \displaystyle 4,900 \div 1,000 = 4.9 kilometers.

Example Question #1 : Rectangles

The length of a rectangle is two times as long as the width. The width is equal to \displaystyle 4 inches. What is the perimeter of the rectangle?

Possible Answers:

\displaystyle \small 16\ in

\displaystyle \small 40\ in

\displaystyle \small 24\ in

 

 

 

 

 

\displaystyle \small 32\ in

Correct answer:

\displaystyle \small 24\ in

 

 

 

 

 

Explanation:

\displaystyle l=2w=2(4)=8

\displaystyle P=l+w+l+w=2l+2w=2(8)+2(4)=16+8=24

Example Question #2 : Rectangles

How many meters of fence are needed to enclose a rectangular field that has a length of 1000 meters and a width of 100 meters?

Possible Answers:

\displaystyle 2000\ meters

\displaystyle 100,000\ meters

\displaystyle 10,000\ meters

\displaystyle 2200\ meters

\displaystyle 1100\ meters

Correct answer:

\displaystyle 2200\ meters

Explanation:

The perimeter of a rectangle is simply the sum of the four sides:

\displaystyle 1000+1000+100+100=2200\; meters

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