GRE Math : How to find the length of the side of a rectangle

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

A rectangle has an area of 48 and a perimeter of 28. What are its dimensions?

Possible Answers:

1 x 48

16 x 3

0.25 x 192

6 x 8

2 x 24

Correct answer:

6 x 8

Explanation:

We can set up our data into the following two equations:

(Area) LH = 48

(Perimeter) 2L + 2H = 28

Solve the area equation for one of the two variables (here, length): L = 48 / H

Place that value for L into ever place you find L in the perimeter equation: 2(48 / H) + 2H = 28; then simplify:

96/H + 2H = 28

Multiply through by H: 96 + 2H² = 28H

Get everything on the same side of the equals sign: 2H² - 28H + 96 = 0

Divide out the common 2: H² - 14H + 48 = 0

Factor: (H - 6) (H - 8) = 0

Either of these multiples can be 0, therefore, consider each one separately:

H - 6 = 0; H = 6

H - 8 = 0; H = 8

Because this is a rectangle, these two dimensions are the height and width. If you choose 6 for the "height" the other perpendicular dimension would be 8 and vice-versa. Therefore, the dimensions are 6 x 8.

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

The length of a rectangle is three times its width, and the perimeter is . What is the width of the rectangle?

Possible Answers:

Correct answer:

Explanation:

For any rectangle, $\dpi{100} \small P=2L+2W$ , where $\dpi{100} \small P=perimeter$ , $\dpi{100} \small L=length$ , and $\dpi{100} \small W=width$ .

In this problem, we are given that $\dpi{100} \small L=3W$ (length is three times the width), so replace $\dpi{100} \small L$ in the perimeter equation with $\dpi{100} \small 3W$ : $\dpi{100} \small P=2(3W)+2W$

Plug in our value for the perimeter, $\dpi{100} \small P$ :

$\dpi{100} \small 16=2(3W)+2W$

Simplify:

$\dpi{100} \small 16=6W+2W$

$\dpi{100} \small 16=8W$

$\dpi{100} \small W=2$

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

The area of a rectangle is $70\:in^2$ . Its perimeter is $38\:in$ . What is the length of its shorter side?

Possible Answers:

$7\:in$

$12\:in$

$8\:in$

$5\:in$

$14\:in$

Correct answer:

$5\:in$

Explanation:

We know that the following two equations hold for rectangles. For area:

A=WL

For perimeter:

P=2W+2L

Now, for our data, we know:

70=WL

38=2W+2L

Now, solve the first equation for one of the variables:

$W=\frac{70}{L}$

Now, substitute this value into the second equation:

$38=2(\frac{70}{L})+2L$

Solve for :

$38=\frac{140}{L}+2L$

Multiply both sides by :

38L=140+2L^2

Solve as a quadratic. Divide through by :

19L=70+L^2

Now, get the equation into standard form:

L^2-19L+70=0

Factor this:

(L-14)(L-5)=0

This means that (or ) would equal either $5\:in$ or $14\:in$ . Therefore, your answer is $5\:in$ .

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GRE Math : How to find the length of the side of a rectangle

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

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