FIO-16

Chapter: SQUARES AND SQUARE ROOTS
Class: 8 (Class 8)
Category: figure_it_out

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Question

Give 5 examples of rectangles whose sidelengths and diagonals are all integers.

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A rectangle's sides and diagonal form a right-angled triangle.

Step 1 — Understanding the rectangle and its diagonal

A rectangle has four sides. Its opposite sides are equal. All its angles are 90 degrees. A diagonal connects opposite corners. It splits the rectangle into two right-angled triangles. We can use the Pythagorean theorem here.

Let $L$ be the length of the rectangle. Let $W$ be its width. Let $D$ be its diagonal. The Pythagorean theorem states:

$L^2 + W^2 = D^2$

We need $L, W, D$ to be whole numbers. These sets of numbers are called Pythagorean triples.

Step 2 — How to find Pythagorean triples

There is a special trick to find these numbers. We can use two counting numbers, let's call them $m$ and $n$. We must choose $m$ to be greater than $n$. The two shorter sides of the right triangle will be $m^2 - n^2$ and $2mn$. The longest side (the diagonal) will be $m^2 + n^2$. We will pick different $m$ and $n$ values to find our examples.

Step 3 — First example

Let us choose $m=2$ and $n=1$.

The first sidelength is $m^2 - n^2$:

$L = 2^2 - 1^2$

$= 4 - 1$

$\boxed{3}$

The second sidelength is $2mn$:

$W = 2 \times 2 \times 1$

$\boxed{4}$

The diagonal is $m^2 + n^2$:

$D = 2^2 + 1^2$

$= 4 + 1$

$\boxed{5}$

So, the first rectangle has sidelengths 3 and 4, and a diagonal of 5.

Step 4 — Second example

Let us choose $m=3$ and $n=2$.

The first sidelength is $m^2 - n^2$:

$L = 3^2 - 2^2$

$= 9 - 4$

$\boxed{5}$

The second sidelength is $2mn$:

$W = 2 \times 3 \times 2$

$\boxed{12}$

The diagonal is $m^2 + n^2$:

$D = 3^2 + 2^2$

$= 9 + 4$

$\boxed{13}$

So, the second rectangle has sidelengths 5 and 12, and a diagonal of 13.

Step 5 — Third example

Let us choose $m=4$ and $n=3$.

The first sidelength is $m^2 - n^2$:

$L = 4^2 - 3^2$

$= 16 - 9$

$\boxed{7}$

The second sidelength is $2mn$:

$W = 2 \times 4 \times 3$

$\boxed{24}$

The diagonal is $m^2 + n^2$:

$D = 4^2 + 3^2$

$= 16 + 9$

$\boxed{25}$

So, the third rectangle has sidelengths 7 and 24, and a diagonal of 25.

Step 6 — Fourth example

Let us choose $m=4$ and $n=1$.

The first sidelength is $m^2 - n^2$:

$L = 4^2 - 1^2$

$= 16 - 1$

$\boxed{15}$

The second sidelength is $2mn$:

$W = 2 \times 4 \times 1$

$\boxed{8}$

The diagonal is $m^2 + n^2$:

$D = 4^2 + 1^2$

$= 16 + 1$

$\boxed{17}$

So, the fourth rectangle has sidelengths 8 and 15, and a diagonal of 17.

Step 7 — Fifth example

Let us choose $m=5$ and $n=4$.

The first sidelength is $m^2 - n^2$:

$L = 5^2 - 4^2$

$= 25 - 16$

$\boxed{9}$

The second sidelength is $2mn$:

$W = 2 \times 5 \times 4$

$\boxed{40}$

The diagonal is $m^2 + n^2$:

$D = 5^2 + 4^2$

$= 25 + 16$

$\boxed{41}$

So, the fifth rectangle has sidelengths 9 and 40, and a diagonal of 41.

Answer

(i) Sidelengths: 3 and 4, Diagonal: 5 (ii) Sidelengths: 5 and 12, Diagonal: 13 (iii) Sidelengths: 7 and 24, Diagonal: 25 (iv) Sidelengths: 8 and 15, Diagonal: 17 (v) Sidelengths: 9 and 40, Diagonal: 41