The side of a rhombus is 10 cm. The smaller diagonal is `(1)/(3)` of the greater diagonal. Find the length of the greater diagonal.

Options

A. `6sqrt(10)cm`
B. `10 sqrt(6)cm`
C. `6 sqrt(5)cm`
D. `5 sqrt(6)cm`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have a rhombus with each side measuring 10 cm. The smaller diagonal is one-third the length of the greater diagonal. We need to find the length of the greater diagonal.

2. **Define Variables:**
Let the length of the greater diagonal \( AC \) be \( x \) cm. Then, the length of the smaller diagonal \( BD \) will be \( \frac{1}{3}x \) cm.

3. **Properties of Rhombus:**
The diagonals of a rhombus bisect each other at right angles. Therefore, if \( O \) is the intersection point of the diagonals:
- \( AO = \frac{x}{2} \) (half of the greater diagonal)
- \( OB = \frac{1}{2} \left(\frac{1}{3}x\right) = \frac{x}{6} \) (half of the smaller diagonal)

4. **Using Pythagoras Theorem:**
In right triangle \( AOB \), we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + OB^2 \]
Given that \( AB = 10 \) cm (the side of the rhombus), we can substitute:
\[ 10^2 = \left(\frac{x}{2}\right)^2 + \left(\frac{x}{6}\right)^2 \]

5. **Expanding the Equation:**
\[ 100 = \frac{x^2}{4} + \frac{x^2}{36} \]
To combine these fractions, find a common denominator. The least common multiple of 4 and 36 is 36:
\[ 100 = \frac{9x^2}{36} + \frac{x^2}{36} = \frac{10x^2}{36} \]

6. **Simplifying the Equation:**
Multiply both sides by 36 to eliminate the fraction:
\[ 3600 = 10x^2 \]
Divide both sides by 10:
\[ x^2 = 360 \]

7. **Finding \( x \):**
Take the square root of both sides:
\[ x = \sqrt{360} = 6\sqrt{10} \]

8. **Conclusion:**
The length of the greater diagonal \( AC \) is \( 6\sqrt{10} \) cm.

### Final Answer:
The length of the greater diagonal is \( 6\sqrt{10} \) cm.

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