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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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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
[Reveal] Spoiler: OA

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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, 08:14
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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, 10:53
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, 17:52
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, 10:43
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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, 17:14
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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Factor table with sign: The useful tool to solve polynomial inequalities
Applying AM-GM inequality into finding extreme/absolute value

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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, 19:20
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] 14 Dec 2017, 19:20
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