• 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

Patricia builds two triangles, each with 30 feet of wood.

This topic has 2 expert replies and 0 member replies

Patricia builds two triangles, each with 30 feet of wood.

Post

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

Difficult



Veritas Prep

Patricia builds two triangles, each with 30 feet of wood. The first triangle ABC is built to maximize the length of the base side. The second triangle DEF is built to maximize the area of the triangle. What is the ratio of the length of the base of triangle ABC to the length of the base of triangle DEF? The lengths of all line segments are integers.

A. 1:1
B. 6:5
C. 5:4
D. 7:5
E. 2:1

OA D

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

Post
Hi All,

We're told that Patricia builds two triangles, EACH with 30 feet of wood. The first triangle ABC is built to maximize the LENGTH of the BASE side. The second triangle DEF is built to maximize the AREA of the triangle. In addition, the lengths of all line segments are INTEGERS. We're asked for the ratio of the length of the base of triangle ABC to the length of the base of triangle DEF. This question has a number of built-in 'logic shortcuts' to it; when combined with the proper Geometry rules, you can actually answer this question without doing much 'math' at all.

To start, it's important to note that all of the side lengths MUST be INTEGERS - and the perimeters of each triangle are 30 - both of these facts limit the number of potential triangles that we can make.

For triangle ABC, we want the largest possible base, meaning that we'll be creating a really long, 'almost flat' triangle. Here, the Triangle Inequality Theorem will be useful. In simple terms, that math rule means that the sum of any two sides of a triangle MUST be greater than the third side... so the two sides that are NOT the base of ABC must sum to a total that is GREATER than the base. Since we're dealing with integers, the base CANNOT be 15... since that would make the sum of the other two sides 15... and 15 is NOT greater than 15. Thus, the base must be 14 and the sum of the other two sides would be 16.

For triangle DEF, we need to find the base... but a triangle has 3 sides and any of them could be the base ... so which side (DE, DF or EF) would be the base? Notice how the answer choices all involve numbers (no variables), so the correct answer implies that the 'base' of DEF cannot possibly be 3 different values. What type of triangle does NOT have 3 different side lengths (or even 2 different side lengths for that matter)? An equilateral triangle! Thus, the three sides of DEF must be 10, 10 and 10. Mathematically-speaking, an equilateral triangle is how you would get the largest area, but that knowledge isn't necessary to answer this question.

The ratio of the two bases is 14:10 --> 7:5

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

_________________
Contact Rich at Rich.C@empowergmat.com

  • +1 Upvote Post
  • Quote
  • Flag
Post
AAPL wrote:
Veritas Prep

Quote:
Patricia builds two triangles, each with 30 feet of wood. The first triangle ABC is built to maximize
the length of the base side. The second triangle DEF is built to maximize the area of the triangle. What is the ratio of the length of the base of triangle ABC to the length of the base of triangle DEF? The lengths of all line segments are integers.

A. 1:1
B. 6:5
C. 5:4
D. 7:5
E. 2:1
$$x + y + z = 30\,\,\,\,\,\left( {\Delta \,\,{\rm{lengths}}\,\,{\rm{positive}}\,\,{\rm{ints}}\,\,{\rm{,}}\,\,\,x\,\,{\rm{base}}} \right)$$
$$?\,\,\, = \,\,\,\,{{x\,\,\,\left( {x\,\,\max } \right)} \over {x\,\,\,\,\left( {{S_\Delta }\max } \right)\,\,\,}}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,\,{7 \over 5}$$
$$\left( * \right)\,\,\,\left\{ \matrix{
x\,\,\,\left( {x\,\,\max } \right)\,\,\,\,::\,\,\,\,\,\,x < y + z = 30 - x\,\,\,\,\,\left( {{\rm{the}}\,\,{\rm{giant}}\,\,{\rm{argument}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,x < 15\,\,\,\,\,\, \Rightarrow \,\,\,\,x\,\,\,\left( {x\,\,\max } \right) = 14\,\,\,\,\,\,\,\left[ {\left( {x,y,z} \right) = \left( {14,8,8} \right)\,\,\,{\rm{viable}}} \right] \hfill \cr
x\,\,\,\,\left( {{S_\Delta }\max } \right)\,\,\,::\,\,\,x = y = z = 10\,\,\,\,\left( {{\rm{perim}}\,\,{\rm{const,}}\,\,{\rm{max}}\,\,{\rm{area}}\,\,\,\, \Rightarrow \,\,\,{\rm{regularity}}} \right) \hfill \cr} \right.\,$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

  • +1 Upvote Post
  • Quote
  • Flag
  • 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
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • 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
  • 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
  • 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
  • Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

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

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • 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

Top First Responders*

1 Jay@ManhattanReview 70 first replies
2 fskilnik@GMATH 50 first replies
3 Brent@GMATPrepNow 50 first replies
4 GMATGuruNY 33 first replies
5 Rich.C@EMPOWERgma... 27 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 fskilnik@GMATH

GMATH Teacher

106 posts
2 image description Brent@GMATPrepNow

GMAT Prep Now Teacher

94 posts
3 image description Max@Math Revolution

Math Revolution

93 posts
4 image description Jay@ManhattanReview

Manhattan Review

82 posts
5 image description GMATGuruNY

The Princeton Review Teacher

77 posts
See More Top Beat The GMAT Experts