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

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Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 23 Nov 2016, 06:55
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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

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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 14 Dec 2017, 20:20
Bunuel wrote:
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


QUICK tip:-
MAX base will be 14 since 15 or more will make it a LINE and NOT a triangle..
so in given answers ONLY D is with a 7 in it..
without any further calculations you can press D as the answer..

otherwise if you are doing untimed, .. The area of equilateral triangle is MAX in given perimeter and similarly a SQUARE in quadrilateral..
REASONING - closer the number, MORE the product

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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 23 Nov 2016, 09:14
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Bunuel wrote:
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


The lengths of 3 sides of triangle ABC is a, b, c where a, b, c are integers and c is the base side.
We have \(c<a+b \implies 2c < a+b+c=30 \implies c<15\). Triangle ABC is built to maximize \(c\), so \(c=14\). The length of base side is 14.

For any triangle that has the same perimeter, equilateral triangle has maximum area. So triangle DEF has 3 sides \(d=e=f=10\). The length of base side is 10.

Hence, the ratio is 14 : 10 = 7 : 5

The answer is D.
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 23 Nov 2016, 11:53
How can you say the equilateral triangle has the maximum area? Pls clarify

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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 23 Nov 2016, 18:52
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ashishahujasham wrote:
How can you say the equilateral triangle has the maximum area? Pls clarify

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I use Heron's formula:

\(S_{ABC}=\sqrt{p(p-a)(p-b)(p-c)}\) where \(p=\frac{a+b+c}{2}=const\)

From inequation: \(xyz \leq ( \frac{x+y+z}{3})^3\) for \(x, y, z>0\). Sign \("="\) occurs \(\iff x=y=z\)

We have \(S_{ABC} \leq \sqrt{p \times \left ( \frac{p-a+p-b+p-c}{3} \right) ^3}= \sqrt{p \times (\frac{p}{3})^3}=\frac{p^2}{3\sqrt{3}}\)

\(maxS_{ABC}=\frac{p^2}{3\sqrt{3}} \iff a=b=c=\frac{2p}{3}\)
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 14 Dec 2017, 11:43
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broall wrote:
Bunuel wrote:
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


The lengths of 3 sides of triangle ABC is a, b, c where a, b, c are integers and c is the base side.
We have \(c<a+b \implies 2c < a+b+c=30 \implies c<15\). Triangle ABC is built to maximize \(c\), so \(c=14\). The length of base side is 14.

For any triangle that has the same perimeter, equilateral triangle has maximum area. So triangle DEF has 3 sides \(d=e=f=10\). The length of base side is 10.

Hence, the ratio is 14 : 10 = 7 : 5

The answer is D.


Where do you get the 2c from? I can't figure out why
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 14 Dec 2017, 18:14
rnz wrote:
broall wrote:
Bunuel wrote:
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


The lengths of 3 sides of triangle ABC is a, b, c where a, b, c are integers and c is the base side.
We have \(c<a+b \implies 2c < a+b+c=30 \implies c<15\). Triangle ABC is built to maximize \(c\), so \(c=14\). The length of base side is 14.

For any triangle that has the same perimeter, equilateral triangle has maximum area. So triangle DEF has 3 sides \(d=e=f=10\). The length of base side is 10.

Hence, the ratio is 14 : 10 = 7 : 5

The answer is D.


Where do you get the 2c from? I can't figure out why


c < a + b => c + c < a + b+ c => 2c < a + b + c
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 08 Oct 2018, 05:35
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Some Simple concepts to solve this tricky Question : ( mug this up )

1. Equilateral Triangle has the largest area of all triangles with same Perimeter.
2. 2a < Perimeter , where a is a side of a Triangle , which basically means that twice of any side of a Triangle cannot be greater than its Perimeter.

Now , coming back to the Question :

Given : Perimeter of both the Triangles = 30.
1. If you want to maximize the area , you will need an Equilateral triangle which makes the side = 10 .
2. If you want to maximize the base , the longest length will be 14 ( because of Rule No. 2 mentioned above ) .

Ratio of these 2 will give you 14/10 = 7:5 ( Answer Option D)
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 12 Jan 2019, 12:49
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.

Since the first part of the ratio is 14, we can eliminate Answers B and C (since they cannot be reduced from 14).

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:

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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 12 Jan 2019, 12:52
chetan2u wrote:
Bunuel wrote:
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


QUICK tip:-
MAX base will be 14 since 15 or more will make it a LINE and NOT a triangle..
so in given answers ONLY D is with a 7 in it..
without any further calculations you can press D as the answer..


Hi chetan2u,

You have to be careful about this type of 'deduction'; 14 can reduce down to 7, 2 or 1.... meaning that while this one piece of information can help you to eliminate Answers B and C, it is not enough to guarantee that the answer is D.

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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 13 Jan 2019, 07:15
Bunuel wrote:
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.
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Re: Patricia builds two triangles, each with 30 feet of wood. The first  [#permalink]

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New post 16 Apr 2019, 18:05
Bunuel wrote:
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


Since triangle ABC maximizes the length of the base, and a side of the triangle must be less than the sum of the other two sides, then the maximum length of the base will be 14, which is 1 less than half of the perimeter.

The triangle with a fixed perimeter that maximizes the area is an equilateral triangle since, if a polygon is given a fixed perimeter, the polygon with the largest area must be regular (note: a “regular” triangle is an equilateral triangle). So a side (or base) of triangle DEF = 10.

Therefore, the ratio of the the two bases = 14/10 = 7/5.

Answer: D
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Re: Patricia builds two triangles, each with 30 feet of wood. The first   [#permalink] 16 Apr 2019, 18:05
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