Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?

Question

Homi · Accepted Answer

Understand the Question

Rectangles with the same ratio of length to width have the same proportion (shape), even if their actual dimensions (sizes) differ.
For a rectangle with a $2:1$ ratio, the length is always twice the width ( $L = 2W$ ).
Comparing different classmates' drawings means checking whether scaling the dimensions changes the overall shape.

Step 1 · Understand the Length-to-Width Ratio

Let $L$ be the length and $W$ be the width of the rectangle.

The given length-to-width ratio is $2:1$ : $\dfrac{L}{W} = \dfrac{2}{1} \implies L = 2 \times W$

For example, if width $W = 4\text{ cm}$ : $L = 2 \times 4\text{ cm} = 8\text{ cm}$ Diagram 1

Step 2 · Compare Rectangles Drawn by Classmates

Classmates following the same $2:1$ ratio might choose different dimensions:

For a width of $3\text{ cm}$ : $L = 2 \times 3\text{ cm} = 6\text{ cm}$
For a width of $6\text{ cm}$ : $L = 2 \times 6\text{ cm} = 12\text{ cm}$

While the actual measurements differ ( $8\text{ cm} \times 4\text{ cm}$ , $6\text{ cm} \times 3\text{ cm}$ , $12\text{ cm} \times 6\text{ cm}$ ), the ratio $\dfrac{L}{W} = 2$ remains identical across all drawings, meaning they all share the exact same shape.

Answer

Yes, they all look the same in shape because they all follow the same $2:1$ length-to-width ratio. Only the size (scale) differs, not the shape.

Common Mistakes

Confusing Size with Shape: Assuming rectangles look different just because one is drawn larger than another. Two figures with proportional corresponding sides are geometrically similar.
Reversing the Ratio: Misinterpreting a $2:1$ length-to-width ratio as width being twice the length instead of length being twice the width.

Question 8

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