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

Question

https://gmatclub.com/forum/download/file.php?id=25491 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 Q3_Img.png

shriramvelamuri · Answer

Hi IMO, answer is C

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.

chetan2u · Answer

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

PareshGmat · Answer

Answer = C. 30 Q3_Img.png 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

Bunuel · Answer

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: https://gmatclub.com/forum/download/file.php?id=25516 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! Q3_ImgSolution.png

AVRonaldo · Answer

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

Hence answer is 30 ( question asks for the approximate one)

user54432 · Answer

I just ballparked,and deduced that if the distance X to AB is 3,the radius would be ~5.Thankfully,the options were merciful.

ScottTargetTestPrep · Answer

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.

Answer: C

jaisonsunny77 · Answer

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

)

bumpbot · Answer

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

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

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