A small, rectangular park has a perimeter of 560 feet and...

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AAPL: Moderator; Posts: 2599; Joined: Sun Oct 29, 2017 2:08 pm; Followed by:2 members

A small, rectangular park has a perimeter of 560 feet and...

by AAPL » Thu Dec 14, 2017 11:31 am

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

A. 19,200
B. 19,600
C. 20,000
D. 20,400
E. 20,800

The OA is A.

I know that the area of a rectangle is A=L*W and its perimeter is P=2*(L+W) but I'm confused with this PS question.

Please, can any expert assist me with it? Thanks in advanced.

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Source: Beat The GMAT — Problem Solving | Thu Dec 14, 2017 11:31 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Thu Dec 14, 2017 11:36 am

A small rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

a/ 19200
b/ 19600
c/ 20000
d/ 20400
e/ 20800

ALWAYS LOOK FOR SPECIAL TRIANGLES.
Draw the rectangle and its diagonal:

Since diagonal AD is a multiple of 5, check whether âˆ†ABD is a multiple of a 3:4:5 triangle.
If each side of a 3:4:5 triangle is multiplied by 40, we get::
(40*3):(40*4):(40*5) = 120:160:200.
The following figure is implied:

Check whether the resulting perimeter for rectangle ABCD is 560:
120+160+120+160 = 560.
Success!
Implication:
For the perimeter of rectangle ABCD to be 560, âˆ†ABD must be a multiple of a 3:4:5 triangle with sides 120, 160 and 200.

Thus:
Area of rectangle ABCD = L * W = 160 * 120 = 19200.

The correct answer is A.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Thu Dec 14, 2017 11:58 am

AAPL wrote:A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

A. 19,200
B. 19,600
C. 20,000
D. 20,400
E. 20,800

Let L and W equal the length and width of the rectangle respectively.

perimeter = 560
So, L + L + W + W = 560
Simplify: 2L + 2W = 560
Divide both sides by 2 to get: L + W = 280

diagonal = 200
The diagonal divides the rectangle into two RIGHT TRIANGLES.
Since we have right triangles, we can apply the Pythagorean Theorem.
We get LÂ² + WÂ² = 200Â²

NOTE: Our goal is to find the value of LW [since this equals the AREA of the rectangle]

If we take L + W = 280 and square both sides we get (L + W)Â² = 280Â²
If we expand this, we get: LÂ² + 2LW + WÂ² = 280Â²
Notice that we have LÂ² + WÂ² "hiding" in this expression.
That is, we get: LÂ² + 2LW + WÂ² = 280Â²

We already know that LÂ² + WÂ² = 200Â², so, we'll take LÂ² + 2LW + WÂ² = 280Â² and replace LÂ² + WÂ² with 200Â² to get:
2LW + 200Â² = 280Â²
So, 2LW = 280Â² - 200Â²
Evaluate: 2LW = 38,400
Solve: LW = [spoiler]19,200 = A[/spoiler]

For extra practice, here's a similar question: https://www.beatthegmat.com/area-of-a-re ... 00119.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Thu Oct 18, 2018 5:24 pm

AAPL wrote:A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

A. 19,200
B. 19,600
C. 20,000
D. 20,400
E. 20,800

We can let L = the length of the park and W = the width of the park. Our goal is to find the area of the park, i.e., the value of LW.

We can create the equations:

Perimeter = 560

2L + 2W = 560

L + W = 280

and

L^2 + W^2 = 200^2

Squaring the first equation, we have:

(L + W)^2 = 280^2

L^2 + W^2 + 2LW = 280^2

Substituting, we have:

200^2 + 2LW = 280^2

2LW = 280^2 - 200^2

2LW = (280 - 200)(280 + 200)

2LW = 80 x 480

LW = 40 x 480 = 19,200

Answer: A

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