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Math Expert V
Joined: 02 Sep 2009
Posts: 64939
Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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8 00:00

Difficulty:   25% (medium)

Question Stats: 76% (02:19) correct 24% (02:35) wrong based on 252 sessions

### HideShow timer Statistics Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

A. 5
B. 10
C. 30
D. 45
E. 75

Kudos for a correct solution.

Attachment: Q3_Img.png [ 22.56 KiB | Viewed 8894 times ]

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Posts: 183
Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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4

step 1: area of rectangle= 8 times the width-> length= 8cm.

Distance from point x to AB= 3; The diagnal of the rectangle passes through teh point X.

Now X divides AB in two halfs- So we can use the pythogorean theorem, We have a side 4, another side 3. The hypotenuse, which is half the diagnal = 5.

Now the diameter of the circle= 10; Cicumference= pi * D= 31.4-> Approximately 30. answer C.

Buneul please correct me if wrong.

Bunuel wrote:
Attachment:
Q3_Img.png
Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

A. 5
B. 10
C. 30
D. 45
E. 75

Kudos for a correct solution.
Math Expert V
Joined: 02 Aug 2009
Posts: 8741
Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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1
Bunuel wrote:
Attachment:
Q3_Img.png
Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

A. 5
B. 10
C. 30
D. 45
E. 75

Kudos for a correct solution.

area =length * width = 8* width..
so length = 8...
it is given that center is 3 units away from length ... In a rectangle, this means that half the width is 3. so width = 6..
dia =hyp of right angle triangle with sides 6 and 8 =10..
circum= 10 pi=31.4 nearly 30... ans C
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Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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Attachment: Q3_Img.png [ 20.51 KiB | Viewed 8115 times ]

Width = 3*2 = 6

Area = 6*8 = 48, so length = 8

Diagonal of rectangle $$= \sqrt{6^2 + 8^2} = 10$$

Radius of circle $$= \frac{10}{2} = 5$$

Circumference $$= 2\pi*5 = 10*3.14 = 31.4$$
Math Expert V
Joined: 02 Sep 2009
Posts: 64939
Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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Bunuel wrote: Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

A. 5
B. 10
C. 30
D. 45
E. 75

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

The correct response is (C). To find the circumference of a circle, we must find the radius. Let’s start with what we know:

The area of a rectangle is lw. Here we are told that lw = 8w. That means the length is 8. We also know that from X to AB =3, so the length of the rectangle must be 6. Let’s re-draw the shape: We can see that a right triangle is formed with the radius AX and half the length and width of the rectangle. Since it’s a classic Pythagorean triplet (3:4:5), we don’t need to use the Pythagorean theorem!

Now that we have the radius, we can plug it into the formula for circumference: C=2πr. C=2∗π∗5. The circumference is approximately 10π, or a number slightly larger than 30.

If you chose (A), you correctly found the radius, but stopped short of finding the circumference. Remember to write down what the question is asking you, especially for this type of multi-step problem, so you don’t stop half-way there!

If you chose (B), the circumference is 10π, but since the question asks for an approximate answer and each choice omits π, you’ll need to multiple 10 by 3 to get the right choice, answer (E).

If you chose (D), this approximation is too large. Even if you were worries that 30 was too small, remember that π = about 3.14. Multiplying is by 10 will only move the decimal one place to the right, giving us 31.4, a number still much closer to (C) than 45.

If you chose (E), you correctly found the radius, but then calculated the approximate area (πr^2) and not the approximate circumference. Review both formulas to make sure you don’t get them mixed up on the GMAT!

Attachment: Q3_ImgSolution.png [ 28.88 KiB | Viewed 8633 times ]

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Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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23a2012 wrote:
Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

5

10

30

45

75

Area of the rectangle = L * B

Given, Area is 8 times its width i.e A = 8 * B --> L= 8

Since rectangle is inscribed within the circle, the diagonal BD is the diameter(2r) of the circle.

Given that the distance between X and AB is 3. (Connect a line from X to AB. This is basically the radius of the circle if you extend till the circle. it bisects AB equally at E)
The distance between B and E is 4 .

So basically it forms a right triangle with two sides 3 & 4. Hence the hypotenuse (i.e radius) is 5 (Pythogorean theorem)

Circumference is 2*pi* r =2 *22/7 * 5 = 31.blah blah

Attachments Q3_Img.png [ 21.4 KiB | Viewed 7900 times ]

Intern  Joined: 28 May 2015
Posts: 10
Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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I just ballparked,and deduced that if the distance X to AB is 3,the radius would be ~5.Thankfully,the options were merciful.
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Re: Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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Bunuel wrote: Rectangle ABCD is inscribed in a circle with center X. If the area of the rectangle is eight times its width, and the distance from X to side AB is three, what is the approximate circumference of the circle?

A. 5
B. 10
C. 30
D. 45
E. 75

Attachment:
Q3_Img.png

Since the area of the rectangle is eight times its width:

8W = L x W

8 = L

We also know that the width is 2(3) = 6.

Using the Pythagorean theorem, we see that the diagonal of the rectangle is d^2 = 8^2 + 6^2, or d = 10. The diagonal of the rectangle is 10, which is also the diameter of the circle.

Thus, we have a circumference of πd, or 10π, which is approximately 30.

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Joined: 05 Jan 2019
Posts: 547
Rectangle ABCD is inscribed in a circle with center X. If the area of  [#permalink]

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I have attached the steps that helped me obtain the solution to this problem. Hope that this helps. (PS: please forgive me for my handwriting )
Attachments

File comment: Detailed steps that I used to arrive at the solution (forgive me for my handwriting) IMG_20200531_183548906.jpg [ 1.21 MiB | Viewed 326 times ] Rectangle ABCD is inscribed in a circle with center X. If the area of   [#permalink] 31 May 2020, 05:13

# Rectangle ABCD is inscribed in a circle with center X. If the area of  