Find the diagonal of a square with sidelength 5 cm.

Question

Homi · Accepted Answer

We can find the diagonal of a square by using the Pythagorean theorem.

Step 1 — Visualizing the Square

Let us draw a square and label its corners. Let the square be ABCD. Each side of the square is 5 cm long. We want to find the length of the diagonal, let us call it BD. The diagonal BD divides the square into two right-angled triangles. Let us consider the triangle BCD. It is a right-angled triangle because all angles in a square are 90 degrees. So, angle C is 90 degrees.

Diagram 1

Step 2 — Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. In triangle BCD, BC and CD are the two shorter sides. BD is the hypotenuse, which is the diagonal we want to find. So, we can write the theorem for triangle BCD.

$BD^2 = BC^2 + CD^2$

We know that $BC = 5 \text{ cm}$ and $CD = 5 \text{ cm}$ . Let us substitute these values into the equation.

$BD^2 = 5^2 + 5^2$

$BD^2 = 25 + 25$

$BD^2 = 50$

To find BD, we need to take the square root of 50.

$BD = \sqrt{50}$

We can simplify the square root of 50. We look for perfect square factors of 50. We know that $50 = 25 \times 2$ . Since $25 = 5^2$ , we can write this as:

$BD = \sqrt{5^2 \times 2}$

$BD = 5 \times \sqrt{2}$

The value of $\sqrt{2}$ is approximately 1.414. Let us use this value to find the approximate length of the diagonal.

$BD = 5 \times 1.414$

$BD = 7.07$

We can round this to one decimal place.

$\boxed{BD \approx 7.1 \text{ cm}}$

Answer

(i) The exact length of the diagonal is $\mathbf{5\sqrt{2} \text{ cm}}$ . (ii) The approximate length of the diagonal is $\mathbf{7.1 \text{ cm}}$ .

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