• NEW! FREE Beat The GMAT Quizzes
    NEW! FREE Beat The GMAT Quizzes
    NEW! FREE Beat The GMAT Quizzes
    Hundreds of Questions Highly Detailed Reporting Expert Explanations TAKE A FREE GMAT QUIZ
  • 7 CATs FREE!
    If you earn 100 Forum Points

    Engage in the Beat The GMAT forums to earn
    100 points for $49 worth of Veritas practice GMATs FREE

    Veritas Prep
    VERITAS PRACTICE GMAT EXAMS
    Earn 10 Points Per Post
    Earn 10 Points Per Thanks
    Earn 10 Points Per Upvote
    REDEEM NOW

In the xy-coordinate system, rectangle ABCD is inscribed

This topic has 1 expert reply and 0 member replies

In the xy-coordinate system, rectangle ABCD is inscribed

Post

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

Difficult



In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x^2 + y^2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y=3x+15, what is the area of rectangle ABCD?

A. 15
B. 30
C. 40
D. 45
E. 50

OA B

Source: GMAT Prep

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post
If a circle has the equation x^2 + y^2 = r^2, then it is a circle centered at the origin, with radius r. So our circle here is a circle of radius 5, centered at (0, 0). So, if a diagonal of the inscribed rectangle lies on the x-axis, the coordinates of its endpoints must be (-5, 0) and (5, 0).

We know the line y = 3x + 15 intersects the circle in two points. If we solve that line's equation along with the circle's equation, we'll find where those two points are. Substituting y = 3x + 15 into the circle's equation, we have

x^2 + (3x + 15)^2 = 25
x^2 + 9x^2 + 90x + 225 = 25
10x^2 + 90x + 200 = 0
x^2 + 9x + 20 = 0
(x + 4)(x + 5) = 0

and the two intersection points are at x = -5 (which we knew already) and x = -4. The y-coordinate of this second point is 3 (which you can find by plugging x = -4 into either the line's or the circle's equation, or you can notice you'll make a 3-4-5 triangle since the radius is 5).

Finally, the diagonal chops the rectangle in half. So the triangle consisting of the points (-5, 0), (5, 0) and (-4, 3) is half of the rectangle. Taking the horizontal edge, along the x-axis, as our base, the length of the base is 10. Then the height is just 3, the y-coordinate of the point above the x-axis. So the area of this triangle is 3*10/2, and that's half the rectangle's area, so the rectangle's area is 30.

_________________
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

  • +1 Upvote Post
  • Quote
  • Flag
  • Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • The Princeton Review
    FREE GMAT Exam
    Know how you'd score today for $0

    Available with Beat the GMAT members only code

    MORE DETAILS
    The Princeton Review

Top First Responders*

1 Ian Stewart 59 first replies
2 Brent@GMATPrepNow 59 first replies
3 Jay@ManhattanReview 53 first replies
4 GMATGuruNY 30 first replies
5 ceilidh.erickson 20 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

Most Active Experts

1 image description Scott@TargetTestPrep

Target Test Prep

188 posts
2 image description Max@Math Revolution

Math Revolution

90 posts
3 image description Brent@GMATPrepNow

GMAT Prep Now Teacher

88 posts
4 image description Ian Stewart

GMATiX Teacher

68 posts
5 image description Jay@ManhattanReview

Manhattan Review

61 posts
See More Top Beat The GMAT Experts