Now suppose we are given a square, and we want to construct a square whose area is half that of the original square. How would you do it?

Question

Homi · Accepted Answer

We can construct a square with half the area of the original square by connecting the midpoints of its sides.

Step 1 — Understanding the Area

Let the side length of the original square be $s$ . The area of this original square is found by multiplying its side length by itself.

$\text{Area of original square} = s \times s$

$= s^2$

We want to construct a new square whose area is half of this.

$\text{Desired area of new square} = \frac{s^2}{2}$

Let the side length of this new square be $s'$ . Its area will be $s'^2$ .

$s'^2 = \frac{s^2}{2}$

To find the side length $s'$ , we take the square root of the desired area.

$s' = \sqrt{\frac{s^2}{2}}$

$\boxed{s' = \frac{s}{\sqrt{2}}}$

This means the side length of the new square must be $s$ divided by $\sqrt{2}$ .

Step 2 — Geometric Construction Method

Let us consider the original square. We can find the midpoint of each of its four sides. A midpoint is the point exactly halfway along a line segment. Let us connect these four midpoints with straight lines. This will form a new figure inside the original square.

Let's check if this new figure is a square and what its area is. Consider one of the corners of the original square. This corner forms a right-angled triangle with two of the midpoints. The two shorter sides (legs) of this triangle are each half the side length of the original square, so they are $s/2$ . The longest side (hypotenuse) of this triangle is one of the sides of our new inner figure. We can use the Pythagorean theorem to find its length. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

$(\text{side of new square})^2 = \left(\frac{s}{2}\right)^2 + \left(\frac{s}{2}\right)^2$

$= \frac{s^2}{4} + \frac{s^2}{4}$

$= \frac{2s^2}{4}$

$= \frac{s^2}{2}$

So, the side length of the new square is $\sqrt{\frac{s^2}{2}}$ .

$\boxed{\text{Side of new square} = \frac{s}{\sqrt{2}}}$

This matches the side length $s'$ we calculated in Step 1. Since all four such triangles are identical, all four sides of the inner figure are equal. Also, the angles of the inner figure are all $90^\circ$ , making it a square. The area of this new square is $\left(\frac{s}{\sqrt{2}}\right)^2 = \frac{s^2}{2}$ , which is half the area of the original square.

Step 3 — Steps for Construction

First, draw the given square. Then, find the midpoint of each of its four sides. You can do this by measuring each side and marking the halfway point. Finally, connect these four midpoints with straight lines. The figure formed in the center is the square with half the area of the original square.

Diagram 1

Answer

To construct a square with half the area of a given square:

(i) Find the midpoint of each of the four sides of the original square. (ii) Connect these four midpoints with straight lines. (iii) The figure formed by connecting the midpoints is the desired square with half the area.

Question 5

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