The perimeter of a rhombus is 40 cm. If one of the diagonals be 12 cm long, what ia the length of the other diagonal ?

Options

A. 12 cm
B. `sqrt(136) cm`
C. 16 cm
D. `sqrt(44) cm`

Answer

To find the length of the other diagonal of the rhombus, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has all four sides equal, and its diagonals bisect each other at right angles.

### Step 2: Calculate the length of one side
Given the perimeter of the rhombus is 40 cm, we can find the length of one side (s) using the formula for the perimeter:
\[ \text{Perimeter} = 4 \times \text{side} \]
Setting this equal to 40 cm:
\[ 4s = 40 \]
Dividing both sides by 4:
\[ s = 10 \text{ cm} \]

### Step 3: Use the properties of the diagonals
Let the diagonals be \(d_1\) and \(d_2\). We know that one diagonal \(d_1 = 12 \text{ cm}\). The diagonals bisect each other, so each half of the diagonals will be:
\[ \frac{d_1}{2} = \frac{12}{2} = 6 \text{ cm} \]
Let the other diagonal \(d_2\) be unknown, so:
\[ \frac{d_2}{2} = x \text{ cm} \]

### Step 4: Apply the Pythagorean theorem
In the right triangle formed by the halves of the diagonals and a side of the rhombus, we can apply the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ 10^2 = 6^2 + x^2 \]
Calculating the squares:
\[ 100 = 36 + x^2 \]

### Step 5: Solve for \(x^2\)
Rearranging the equation to isolate \(x^2\):
\[ x^2 = 100 - 36 \]
\[ x^2 = 64 \]

### Step 6: Find \(x\)
Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \text{ cm} \]

### Step 7: Calculate the length of the other diagonal
Since \(x\) is half of \(d_2\):
\[ d_2 = 2x = 2 \times 8 = 16 \text{ cm} \]

### Final Answer
The length of the other diagonal \(d_2\) is **16 cm**.

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