The perimeter of a rhombus is 52 cm. If its one diagonal is 10 cm, find using Heron's formula the area of rhombus.

Answer

To find the area of the rhombus using Heron's formula, follow these steps:

### Step 1: Determine the length of one side of the rhombus.
Given that the perimeter of the rhombus is 52 cm, and since all sides of a rhombus are equal, we can find the length of one side (let's call it \( a \)) as follows:

\[ \text{Perimeter} = 4a = 52 \text{ cm} \]

Dividing both sides by 4 gives:

\[ a = \frac{52}{4} = 13 \text{ cm} \]

### Step 2: Identify the lengths of the sides of triangle ABC.
In the rhombus, one diagonal is given as 10 cm. We can denote the vertices of the rhombus as A, B, C, and D. The diagonals intersect at point O, which divides each diagonal into two equal parts. Therefore, the lengths of the sides of triangle ABC are:

- \( AB = a = 13 \text{ cm} \)
- \( BC = a = 13 \text{ cm} \)
- \( AC = 10 \text{ cm} \)

### Step 3: Calculate the semi-perimeter (s) of triangle ABC.
The semi-perimeter \( s \) is calculated using the formula:

\[ s = \frac{AB + BC + AC}{2} = \frac{13 + 13 + 10}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Step 4: Apply Heron's formula to find the area of triangle ABC.
Heron's formula for the area \( A \) of a triangle is given by:

\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values we have:

- \( s = 18 \text{ cm} \)
- \( a = 13 \text{ cm} \)
- \( b = 13 \text{ cm} \)
- \( c = 10 \text{ cm} \)

Calculating each term:

\[ s - a = 18 - 13 = 5 \]
\[ s - b = 18 - 13 = 5 \]
\[ s - c = 18 - 10 = 8 \]

Now substituting these values into Heron's formula:

\[ A = \sqrt{18 \times 5 \times 5 \times 8} \]

### Step 5: Simplify the expression.
Calculating inside the square root:

\[ A = \sqrt{18 \times 25 \times 8} \]

Calculating \( 18 \times 25 = 450 \) and \( 450 \times 8 = 3600 \):

\[ A = \sqrt{3600} \]

Calculating the square root:

\[ A = 60 \text{ cm}^2 \]

### Step 6: Calculate the area of the rhombus.
Since the area of the rhombus is twice the area of triangle ABC, we have:

\[ \text{Area of rhombus} = 2 \times A = 2 \times 60 = 120 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is \( 120 \text{ cm}^2 \).

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