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

Question

Homi · Accepted Answer

Understand the Question

The diagonals of a rhombus are perpendicular bisectors of each other.
When the diagonals intersect, they divide the rhombus into four congruent right-angled triangles.
The legs of each right-angled triangle are half the lengths of the diagonals, and the hypotenuse is the side length ( $a$ ) of the rhombus.
We can use the Pythagoras theorem to find the side length: $a^2 = \left(\dfrac{d_1}{2}\right)^2 + \left(\dfrac{d_2}{2}\right)^2$ .

Step 1 · Find the Lengths of the Half-Diagonals

Let the lengths of the diagonals be $d_1 = 24 \text{ units}$ and $d_2 = 70 \text{ units}$ . Diagram 1

Since the diagonals of a rhombus bisect each other at right angles ( $90^\circ$ ), the lengths of the half-diagonals $h_1$ and $h_2$ are:

\begin{aligned} h_1 &= \dfrac{d_1}{2} = \dfrac{24}{2} = 12 \text{ units} \\[0.6em] h_2 &= \dfrac{d_2}{2} = \dfrac{70}{2} = 35 \text{ units} \end{aligned}

Step 2 · Calculate the Sidelength using Pythagoras Theorem

Let $a$ be the sidelength of the rhombus. In the right-angled triangle formed by the half-diagonals:

$a^2 = h_1^2 + h_2^2$

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

Calculate the squares:

\begin{aligned} 12^2 &= 144 \\[0.6em] 35^2 &= 1225 \end{aligned}

Add the squares:

\begin{aligned} a^2 &= 144 + 1225 \\[0.6em] &= 1369 \end{aligned}

Take the square root to find $a$ : $a = \sqrt{1369} = 37 \text{ units}$

Answer

37 units

Common Mistakes

Using full diagonals in Pythagoras theorem: Using $24^2 + 70^2$ instead of halving the diagonals first, which results in finding double the side length.
Assuming diagonals are equal: A rhombus has equal sides, but its diagonals are usually of different lengths unless it is a square.

Question 13