Perimeter of rectangle and equilateral triangle. Please help

[nikhilgmat31]

If the perimeter of a rectangle is less than the perimeter of equilateral triangle then which one of the following is true?

I. The larger side of the rectangle must be greater than the side of the triangle.

II. The smaller side of the rectangle must be less than the side of the triangle.

III. The small side of the rectangle may be equal to the side of the triangle.

IV. The triangle and the rectangle can share one side.

[theCEO]

If the perimeter of a rectangle is less than the perimeter of equilateral triangle then which one of the following is true?

I. The larger side of the rectangle must be greater than the side of the triangle.

II. The smaller side of the rectangle must be less than the side of the triangle.

III. The small side of the rectangle may be equal to the side of the triangle.

IV. The triangle and the rectangle can share one side.

Here is the start:

perimeter of rectangle < perimeter of triangle
by assigning variables to the sides
2ra+2rb <3t
2(ra+rb)<3t
ra+rb < 1.5t

Process of elimination
I. The larger side of the rectangle must be greater than the side of the triangle.
ra+rb < 1.5t
2 +1 < 1.5*4=6 -> FALSE

III. The small side of the rectangle may be equal to the side of the triangle.
ra+rb < 1.5t
2 +1 < 1.5*1=1.5 - > FALSE

[nikhilgmat31]

Options III and IV seems same to me

[nikhilgmat31]

Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

[theCEO]

Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

Thanks for the question nikhilgmat31

II. The smaller side of the rectangle must be less than the side of the triangle
ra+rb < 1.5t

if we set both sides to being equal,
ra+ra = 1.5t
2 ra = 1.5t
ra = 0.75 t
this shows that one of the sides of rect. is always going to be smaller than side of triangle

[nikhilgmat31]

so II is the correct answer ?

[theCEO]

Thanks My CEO

We can eliminate IV as triangle side can't be any side of rectangle.

Rectangle = ra=1 rb=2

Triangle side t=1 or 2

1+2 <1.5 * 1 ==> 3 <1.5 not possible
1+2 <1.5 * 2 ==> 3<3 not possible

so IV is not possible.

Answer is II

I think IV is possible

IV. The triangle and the rectangle can share one side
ra+rb < 1.5t
1+0.4<1.5*1 -> true

[nikhilgmat31]

introduces confusion.

I thought we finished question

[nikhilgmat31]

1 & 0.4 is basically same as Option II. side of triangle is 1


II. The smaller side of the rectangle must be less than the side of the triangle.

so answer is again II

[Ian Stewart]

Say the equilateral triangle has sides of length x. Its perimeter is 3x.

Say the rectangle has dimensions L and W, where W < L. Its perimeter is 2W + 2L. If we've assumed W < L, then the rectangle's perimeter is thus greater than 2W + 2W = 4W.

We know that the triangle's perimeter exceeds the rectangle's perimeter, which, as we just saw, exceeds 4W. So 3x > 4W, and x > (4/3)W. Since W is positive, (4/3)W is definitely bigger than W, so if x > (4/3)W, then x > W is true, which is what II says.

It is also possible for the rectangle and triangle to share one side, though that side needs to be the length of the rectangle. If, say the sides of the triangle are 1, 1, 1, then the length of the rectangle can certainly be 1. The perimeter of the rectangle will then be less than that of the triangle as long as the width of the rectangle is less than 0.5. So IV can also be true.

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Hi nikhilgmat31,

What is the source of this question? Is it from a GMAT book or a "math" book?

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

[nikhilgmat31]

I took a mock test from egamt.com & got this question there.