If perimeter of a Rhombus is 150 cm. If one diagonal is 25 cm find other diagonal and area :

Answer

To solve the problem step by step, we will follow the given information about the rhombus and apply relevant geometric formulas.

### Step 1: Find the length of one side of the rhombus.
The perimeter of a rhombus is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
Given that the perimeter is 150 cm, we can find the length of one side (let's denote it as \( a \)):
\[ 4a = 150 \]
\[ a = \frac{150}{4} = 37.5 \text{ cm} \]

### Step 2: Identify the diagonals.
Let the diagonals be \( AC \) and \( BD \). We know that \( BD = 25 \) cm. The diagonals of a rhombus bisect each other at right angles. Therefore, we can denote:
\[ \frac{BD}{2} = \frac{25}{2} = 12.5 \text{ cm} \]
Let \( AC = 2x \), where \( x \) is half of diagonal \( AC \).

### Step 3: Apply the Pythagorean theorem.
In the right triangle formed by half of each diagonal and the side of the rhombus, we can apply the Pythagorean theorem:
\[ a^2 = \left(\frac{BD}{2}\right)^2 + \left(\frac{AC}{2}\right)^2 \]
Substituting the known values:
\[ (37.5)^2 = (12.5)^2 + x^2 \]
Calculating the squares:
\[ 1406.25 = 156.25 + x^2 \]
Now, isolate \( x^2 \):
\[ x^2 = 1406.25 - 156.25 = 1250 \]
Taking the square root:
\[ x = \sqrt{1250} = 25\sqrt{2} \text{ cm} \]
Thus, the full diagonal \( AC \) is:
\[ AC = 2x = 2 \times 25\sqrt{2} = 50\sqrt{2} \text{ cm} \]

### Step 4: Calculate the area of the rhombus.
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Substituting the values:
\[ A = \frac{1}{2} \times 25 \times 50\sqrt{2} \]
Calculating this:
\[ A = \frac{1}{2} \times 1250\sqrt{2} = 625\sqrt{2} \text{ cm}^2 \]

### Final Results:
- The other diagonal \( AC = 50\sqrt{2} \text{ cm} \)
- The area of the rhombus \( A = 625\sqrt{2} \text{ cm}^2 \)

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