Understanding the Rhombus Problem: Finding the Other Diagonal

The question asks us to find the length of the second diagonal of a rhombus given the length of one side and the length of one diagonal.

Here's the information provided:

• Length of one side of the rhombus = 13 cm
• Length of one diagonal = 10 cm

We need to find the length of the other diagonal.

Key Properties of a Rhombus

To solve this problem, we need to remember some important properties of a rhombus:

• All four sides of a rhombus are equal in length.
• The diagonals of a rhombus bisect each other (cut each other in half).
• The diagonals of a rhombus intersect each other at a right angle ($90^{\circ}$).

These properties mean that the two diagonals divide the rhombus into four congruent right-angled triangles. The sides of each right-angled triangle are half the length of each diagonal, and the hypotenuse is the side of the rhombus.

Applying the Pythagorean Theorem

Let the side length of the rhombus be $s$. We are given $s = 13$ cm.

Let the two diagonals be $d_1$ and $d_2$. We are given one diagonal, let's say $d_1 = 10$ cm.

The diagonals bisect each other, so half of the first diagonal is $\frac{d_1}{2} = \frac{10}{2} = 5$ cm.

Let the other diagonal be $d_2$. Half of the other diagonal is $\frac{d_2}{2}$.

In one of the four right-angled triangles formed by the diagonals, the lengths of the two shorter sides (legs) are $\frac{d_1}{2}$ and $\frac{d_2}{2}$, and the length of the longest side (hypotenuse) is $s$.

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, we have the relationship:

$\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2$

Now, substitute the known values:

$(5)^2 + \left(\frac{d_2}{2}\right)^2 = (13)^2$

Calculate the squares:

$25 + \left(\frac{d_2}{2}\right)^2 = 169$

Subtract 25 from both sides to isolate the term with $d_2$:

$\left(\frac{d_2}{2}\right)^2 = 169 - 25$

$\left(\frac{d_2}{2}\right)^2 = 144$

Take the square root of both sides to find $\frac{d_2}{2}$:

$\frac{d_2}{2} = \sqrt{144}$

$\frac{d_2}{2} = 12$

Finally, multiply by 2 to find the full length of the other diagonal $d_2$:

$d_2 = 12 \times 2$

$d_2 = 24$ cm

Conclusion

The length of the other diagonal is 24 cm.

Revision Table: Rhombus Properties

Additional Information: Rhombus Calculations

The relationship between the side ($s$) and the diagonals ($d_1$, $d_2$) of a rhombus is a direct application of the Pythagorean theorem. Because the diagonals form four right-angled triangles, we can always use the formula $s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$. This can also be written as $4s^2 = d_1^2 + d_2^2$. This formula is useful for finding a side, a diagonal, or verifying the dimensions of a rhombus. Knowing any two of the three values ($s$, $d_1$, $d_2$) allows you to calculate the third.