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Figures X and Y above (OG2016)

This topic has 2 expert replies and 1 member reply
boomgoesthegmat Senior | Next Rank: 100 Posts Default Avatar
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Figures X and Y above (OG2016)

Post Fri May 06, 2016 12:37 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])


    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

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    OptimusPrep Master | Next Rank: 500 Posts
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    Post Sat May 07, 2016 7:55 pm
    boomgoesthegmat wrote:


    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

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    Post Thu May 12, 2016 12:23 am
    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.

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    Post Mon Sep 11, 2017 12:23 pm
    boomgoesthegmat wrote:


    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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