Solving for the Rhombus Diagonal Length

The question asks us to find the length of the other diagonal of a rhombus, given its perimeter and the length of one diagonal.

Understanding Rhombus Properties

A rhombus is a quadrilateral with four sides of equal length. Its diagonals have two important properties related to this problem:

• The diagonals of a rhombus bisect each other. This means they cut each other in half at their intersection point.
• The diagonals intersect at a right angle (90 degrees).

These properties mean that the diagonals divide the rhombus into four congruent right-angled triangles. The sides of these triangles are:

• The hypotenuse: A side of the rhombus.
• The two legs: Half the length of each diagonal.

Step-by-Step Calculation

1. Find the side length of the rhombus

The perimeter of the rhombus is given as 26 cm.

Since all four sides of a rhombus are equal in length, the length of one side is:

Side length = \( \frac{\text{Perimeter}}{4} \)

Side length = \( \frac{26 \text{ cm}}{4} = 6.5 \text{ cm} \)

So, each side of the rhombus is 6.5 cm long.

2. Determine half the length of the known diagonal

One diagonal of the rhombus is given as 5 cm.

Since the diagonals bisect each other, half the length of this diagonal is:

Half diagonal 1 = \( \frac{5 \text{ cm}}{2} = 2.5 \text{ cm} \)

3. Use the Pythagorean theorem

Consider one of the four right-angled triangles formed by the diagonals. Let the length of the other diagonal be \(d_2\).

The sides of this right-angled triangle are:

• Hypotenuse = Side length = 6.5 cm
• One leg = Half of the known diagonal = 2.5 cm
• Other leg = Half of the unknown diagonal = \( \frac{d_2}{2} \)

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:

\( (\text{Hypotenuse})^2 = (\text{Leg 1})^2 + (\text{Leg 2})^2 \)

\( (\text{Side length})^2 = (\text{Half diagonal 1})^2 + (\text{Half diagonal 2})^2 \)

Substitute the known values:

\( (6.5)^2 = (2.5)^2 + \left(\frac{d_2}{2}\right)^2 \)

Calculate the squares:

\( 42.25 = 6.25 + \left(\frac{d_2}{2}\right)^2 \)

Subtract 6.25 from both sides:

\( 42.25 - 6.25 = \left(\frac{d_2}{2}\right)^2 \)

\( 36 = \left(\frac{d_2}{2}\right)^2 \)

Take the square root of both sides:

\( \sqrt{36} = \sqrt{\left(\frac{d_2}{2}\right)^2} \)

\( 6 = \frac{d_2}{2} \)

4. Find the length of the other diagonal

Multiply by 2 to find the full length of the other diagonal:

\( d_2 = 6 \times 2 \)

\( d_2 = 12 \text{ cm} \)

Thus, the length of the other diagonal is 12 cm.

Summary of Calculations

Revision Table: Rhombus Formulas

Additional Information: Properties of Quadrilaterals

Understanding the properties of different quadrilaterals helps in solving geometry problems. Here are some key types:

• Square: All sides equal, all angles 90°, diagonals equal and bisect each other at 90°.
• Rectangle: Opposite sides equal, all angles 90°, diagonals equal and bisect each other.
• Rhombus: All sides equal, opposite angles equal, diagonals bisect each other at 90°.
• Parallelogram: Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.
• Trapezium (Trapezoid): At least one pair of parallel sides.
• Kite: Two pairs of equal-length sides adjacent to each other, diagonals are perpendicular, one diagonal bisects the other.

The problem specifically uses the right-angle intersection and bisection property of rhombus diagonals along with the equal side lengths defined by the perimeter.