Figures X and Y above (OG2016)

[boomgoesthegmat]

Figures X and Y above show how eight identical triangular pieces of cardboard were used to form a square and a rectangle, respectively. What is the ratio of the perimeter of X to the perimeter of Y?
A) 2 : 3
B) sqrt2 : 2
C) 2sqrt2 : 3
D) 1 : 1
E) sqrt2 : 1

OA: C

[OptimusPrep]

Figures X and Y above show how eight identical triangular pieces of cardboard were used to form a square and a rectangle, respectively. What is the ratio of the perimeter of X to the perimeter of Y?
A) 2 : 3
B) sqrt2 : 2
C) 2sqrt2 : 3
D) 1 : 1
E) sqrt2 : 1


OA: C

The triangles given have to be isoceles triangles to make the square and rectangle
Assume the side of the triangle = x
The hypotenuse = √2x

Hence Side of the square = √2x
Perimeter = 4*√2x

Length of the rectangle = 2x
Breadth of the rectangle = x

Perimeter = 2 (x + 2x) = 6x

Ratio of perimeter of square to rectangle = 4*√2x : 6x = 2√2 : 3

Correct Option: C

[Matt@VeritasPrep]

Suppose each side of the square has length 1. Its perimeter = 4.

The side of the square is a diagonal to the midpoint of y. That triangle has angles of 45°, so each leg has length 1/√2, or √2/2.

The perimeter of that rectangle is 6 * √2/2, or 3√2.

The ratio is thus 4 : 3√2. Dividing both parts of the ratio by √2 gives 2√2 : 3, or C.

[Brent@GMATPrepNow]

Figures X and Y above show how eight identical triangular pieces of cardboard were used to form a square and a rectangle, respectively. What is the ratio of the perimeter of X to the perimeter of Y?
A) 2 : 3
B) √2 : 2
C) 2√2 : 3
D) 1 : 1
E) √2 : 1

OA: C

Let's start by gathering more information about these eight identical triangles.
First notice that we have 4 equal angles meeting at a single point.


So, each angle must be 90°


Now examine the red triangle below.


The red triangle is an isosceles triangle since all 4 sides of a square are equal.
The two equal angles must add to 90°
So, each angle must be 45°


Using similar logic, we can conclude that all of the eight triangles are 45-45-90 special right triangles.


Now let's examine what happens if we examine one particular 45-45-90 special right triangle, which has sides of length 1, 1 and √2


Use those lengths for all eight identical triangles in the 2 diagrams we get the following:


At this point, we can calculate the perimeters.
Perimeter of X = √2 + √2 + √2 + √2 = 4√2
Perimeter of Y = 1 + 1 + 1 + 1 + 1 + 1 = 6

So, perimeter of X : perimeter of Y = 4√2 : 6
Divide both sides by 2 to get the equivalent ratio 2√2 : 3

Answer: C

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Hi All,

This prompt shows us two shapes (a square and a rectangle) that were made up using 8 IDENTICAL triangles. We’re asked for the ratio of the PERIMETER of the square to the PERIMETER of the rectangle. It’s worth noting that the 4 sides of the square are the HYPOTENEUSE of the triangles and the 6 segments that make up the perimeter of the rectangle are the LEGS of the triangles. This question can be solved in a couple of different ways, including by TESTing VALUES.

IF… the triangles are Isosceles triangles with sides of 3, 3 and 3Root2, then…

The perimeter of the square is 4(3Root2) = 12Root2

The perimeter of the rectangle is 6(3) = 18

Thus, the ratio is 12Root2 : 18, which can be reduced by dividing both pieces by 6… This gives us a ratio of 2Root2 : 3

Final Answer: C

GMAT Assassins aren’t born, they’re made,
Rich