What if we wish to combine two squares of 'different' sizes to make a large square whose area is the sum of the two smaller squares?

Question

Homi · Accepted Answer

The problem describes a relationship between the areas of squares, which can be solved using the Pythagorean theorem.

Step 1 — Defining the square areas

Let us consider two squares of different sizes. Let the side length of the first square be $a$ . Its area will be $a \times a$ , which is $a^2$ . Let the side length of the second square be $b$ . Its area will be $b \times b$ , which is $b^2$ . We want to combine them to make a new, larger square. Let the side length of this new large square be $c$ . Its area will be $c \times c$ , which is $c^2$ . The problem states that the area of the large square is the sum of the areas of the two smaller squares. So, we can write this as an equation:

$c^2 = a^2 + b^2$

$\boxed{\text{Area of large square} = \text{Area of small square 1} + \text{Area of small square 2}}$

Diagram 1

Step 2 — Applying the Pythagorean Theorem

The equation $c^2 = a^2 + b^2$ is very special in geometry. It is the famous Pythagorean theorem. This theorem applies to right-angled triangles. A right-angled triangle has one angle that is exactly 90 degrees. The longest side of a right-angled triangle is called the hypotenuse. The other two sides are called legs. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. If we have a right-angled triangle with legs of length $a$ and $b$ , and its hypotenuse is $c$ , then:

$a^2 + b^2 = c^2$

This is exactly the same relationship we found for the areas of our squares. So, to find the side length $c$ of the new large square, we need to find the hypotenuse of a right-angled triangle. The legs of this triangle must be the side lengths of the two smaller squares, $a$ and $b$ .

$\boxed{\text{The side 'c' is the hypotenuse of a right triangle with legs 'a' and 'b'}}$

Diagram 2

Step 3 — Constructing the large square

First, draw a straight line segment of length $a$ . At one end of this segment, draw another line segment of length $b$ . Make sure this second segment is at a 90-degree angle to the first segment. Now, connect the open ends of these two segments. This connecting line will be the hypotenuse of the right-angled triangle. Let its length be $c$ . This length $c$ is the side length of the new large square we want to create. Finally, construct a square using this length $c$ as its side. The area of this new square will be $c^2$ , which is $a^2 + b^2$ .

$\boxed{\text{The new square has side length equal to the hypotenuse}}$

Diagram 3

Answer

(i) The side length of the new large square will be the hypotenuse of a right-angled triangle. (ii) The legs of this right-angled triangle must be the side lengths of the two smaller squares. (iii) This method uses the Pythagorean theorem to find the side length of the combined square.

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