Find the sidelength of a rhombus whose diagonals are of length 24 units and 70 units.

Question

Homi · Accepted Answer

We can find the side length of the rhombus by using its properties and the Pythagorean theorem.

Step 1 — Finding the lengths of half-diagonals

A rhombus is a special type of quadrilateral where all four sides are equal. Its diagonals cut each other exactly in half. They also meet at a perfect right angle (90 degrees).

Let the two diagonals of the rhombus be $d_1$ and $d_2$ . We are given their lengths as 24 units and 70 units. So, $d_1 = 24$ units and $d_2 = 70$ units.

When the diagonals bisect each other, they form four smaller right-angled triangles inside the rhombus. The legs of these triangles are half the lengths of the diagonals.

Let the half-length of the first diagonal be $h_1$ . $h_1 = \frac{d_1}{2}$ $h_1 = \frac{24}{2}$

$\boxed{h_1 = 12 \text{ units}}$

Let the half-length of the second diagonal be $h_2$ . $h_2 = \frac{d_2}{2}$ $h_2 = \frac{70}{2}$

$\boxed{h_2 = 35 \text{ units}}$

Diagram 1

Step 2 — Using the Pythagorean Theorem

Each of the four triangles formed by the diagonals is a right-angled triangle. The sides of the rhombus are the hypotenuses of these triangles.

Let 'a' be the sidelength of the rhombus. In one of these right-angled triangles (for example, triangle AOD), the two shorter sides (legs) are $h_1$ and $h_2$ . The longest side (hypotenuse) is 'a'.

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. So, $a^2 = h_1^2 + h_2^2$ .

$a^2 = 12^2 + 35^2$

First, we calculate the squares of the half-diagonals. $12^2 = 12 \times 12 = 144$ $35^2 = 35 \times 35 = 1225$

Now, we add these values together. $a^2 = 144 + 1225$ $a^2 = 1369$

To find 'a', we need to calculate the square root of 1369. $a = \sqrt{1369}$

$\boxed{a = 37 \text{ units}}$

The sidelength of the rhombus is 37 units.

Answer

The sidelength of the rhombus is 37 units.

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