FIO-18

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

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Question

(i) Using the dots of a grid as the vertices, can you create a square that has an area of (a) 2 sq. units, (b) 3 sq. units, (c) 4 sq.units, and (d) 5 sq. unit?

(ii) Suppose the grid extends indefinitely. What are the possible integer-valued areas of squares you can create in this manner?

Question diagram(s):

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The area of any square whose corners are dots on a grid must be a number that can be written as the sum of two perfect squares.

Step 1 — Understanding Grid Squares

Let us imagine our grid dots are at whole number coordinates, like (0,0), (1,0), (0,1), and so on. The distance between two adjacent dots, either horizontally or vertically, is 1 unit.

If we draw a line segment connecting two dots, say $(x_1, y_1)$ and $(x_2, y_2)$, we can find its length using the Pythagorean theorem. This theorem tells us that for a right-angled triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Let the horizontal distance between the two dots be $a = |x_2 - x_1|$. Let the vertical distance between the two dots be $b = |y_2 - y_1|$. Since $x_1, y_1, x_2, y_2$ are whole numbers, $a$ and $b$ will also be whole numbers. If $s$ is the length of the line segment (which will be the side of our square), then:

$s^2 = a^2 + b^2$

The area of a square is its side length multiplied by itself, which is $s \times s = s^2$. So, the area of any square formed by grid dots must be equal to $a^2 + b^2$, where $a$ and $b$ are whole numbers.

Step 2 — Area of 2 square units

We want to know if we can make a square with an area of 2 square units. This means we need to find two whole numbers, $a$ and $b$, such that $a^2 + b^2 = 2$.

Let us try some whole numbers: If $a = 1$ and $b = 1$:

$1^2 + 1^2 = 1 + 1$

$= 2$

Yes, we found $a=1$ and $b=1$. So, a square with an area of 2 square units is possible. To draw this square, we can pick four dots. Let's start with a dot, say A, in the second column from the left and the first row from the bottom. From A, move 1 dot to the right and 1 dot up to find dot B. From B, move 1 dot to the left and 1 dot up to find dot C. From C, move 1 dot to the left and 1 dot down to find dot D. Joining A, B, C, and D forms a square with an area of 2 square units.

Step 3 — Area of 3 square units

Now, let us check if a square with an area of 3 square units is possible. We need to find two whole numbers, $a$ and $b$, such that $a^2 + b^2 = 3$.

Let us try different whole numbers for $a$ and $b$: If $a = 0$, then $0^2 + b^2 = 3$, which means $b^2 = 3$. There is no whole number $b$ whose square is 3. If $a = 1$, then $1^2 + b^2 = 3$, which means $1 + b^2 = 3$, so $b^2 = 2$. There is no whole number $b$ whose square is 2. If $a = 2$, then $2^2 + b^2 = 3$, which means $4 + b^2 = 3$. This would make $b^2 = -1$, which is not possible for any real number, let alone a whole number. Any larger value for $a$ would make $a^2$ even bigger than 3.

Since we cannot find any whole numbers $a$ and $b$ for which $a^2 + b^2 = 3$, a square with an area of 3 square units is not possible.

Step 4 — Area of 4 square units

Next, let us see if we can make a square with an area of 4 square units. We need to find two whole numbers, $a$ and $b$, such that $a^2 + b^2 = 4$.

Let us try some whole numbers: If $a = 2$ and $b = 0$:

$2^2 + 0^2 = 4 + 0$

$= 4$

Yes, we found $a=2$ and $b=0$. So, a square with an area of 4 square units is possible. To draw this square, we can pick four dots. Let's start with dot A at the bottom-left corner of the grid. From A, move 2 dots to the right to find dot B. From B, move 2 dots up to find dot C. From C, move 2 dots to the left to find dot D. Joining A, B, C, and D forms a square with an area of 4 square units.

Step 5 — Area of 5 square units

Finally, let us check for a square with an area of 5 square units. We need to find two whole numbers, $a$ and $b$, such that $a^2 + b^2 = 5$.

Let us try some whole numbers: If $a = 1$ and $b = 2$:

$1^2 + 2^2 = 1 + 4$

$= 5$

Yes, we found $a=1$ and $b=2$. So, a square with an area of 5 square units is possible. To draw this square, we can pick four dots. Let's start with dot A in the first column from the left and the third row from the bottom. From A, move 2 dots to the right and 1 dot up to find dot B. From B, move 1 dot to the right and 2 dots down to find dot C. From C, move 2 dots to the left and 1 dot down to find dot D. Joining A, B, C, and D forms a square with an area of 5 square units.

Step 6 — Possible Integer Areas

We learned that the area of any square whose corners are grid dots must be equal to $a^2 + b^2$, where $a$ and $b$ are whole numbers representing the horizontal and vertical distances between two adjacent corners of the square.

So, to find all possible integer-valued areas, we just need to find all numbers that can be written as the sum of two whole number squares. Let's list some of these: $0^2 + 0^2 = 0$ (This would be a single dot, not a square with area) $1^2 + 0^2 = 1$ $1^2 + 1^2 = 2$ $2^2 + 0^2 = 4$ $2^2 + 1^2 = 5$ $2^2 + 2^2 = 8$ $3^2 + 0^2 = 9$ $3^2 + 1^2 = 10$ $3^2 + 2^2 = 13$ $3^2 + 3^2 = 18$ $4^2 + 0^2 = 16$ $4^2 + 1^2 = 17$ $4^2 + 2^2 = 20$

The possible integer-valued areas are numbers that can be expressed as the sum of two whole number squares. These are:

$\boxed{1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, \text{ and so on.}}$

Answer

(i) (a) A square with area 2 sq. units is possible. (i) (b) A square with area 3 sq. units is not possible. (i) (c) A square with area 4 sq. units is possible. (i) (d) A square with area 5 sq. units is possible. (ii) The possible integer-valued areas are numbers that can be written as the sum of two whole number squares ($a^2 + b^2$, where $a$ and $b$ are whole numbers). Examples include 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, and so on.