BL and CN are the two medians of the `DeltaABC` where `angleA=90^(@)`. If `BC=5cm,BL=(3sqrt5)/(2)`cm then find the value of CN.

Options

A. `3sqrt5`
B. `sqrt5`
C. `2sqrt5`
D. `5sqrt3`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given values
We have triangle ABC with angle A = 90 degrees. The lengths given are:
- BC = 5 cm
- Median BL = (3√5)/2 cm

### Step 2: Set up the triangle
Since angle A is 90 degrees, we can use the properties of right triangles. Let:
- AB = 2x
- AC = 2y
- BC = 5 cm (as given)

### Step 3: Apply Pythagorean theorem
Using the Pythagorean theorem in triangle ABC:
\[ AB^2 + AC^2 = BC^2 \]
Substituting the values:
\[ (2x)^2 + (2y)^2 = 5^2 \]
This simplifies to:
\[ 4x^2 + 4y^2 = 25 \]
Dividing by 4 gives us:
\[ x^2 + y^2 = \frac{25}{4} \] (Equation 1)

### Step 4: Set up the median
The median BL divides side AC into two equal parts at point L. Therefore, AL = LC = y.

Using the Pythagorean theorem in triangle ABL:
\[ AB^2 + AL^2 = BL^2 \]
Substituting the values:
\[ (2x)^2 + y^2 = \left(\frac{3\sqrt{5}}{2}\right)^2 \]
This simplifies to:
\[ 4x^2 + y^2 = \frac{45}{4} \] (Equation 2)

### Step 5: Solve the equations
Now we have two equations:
1. \( 4x^2 + 4y^2 = 25 \) (Equation 1)
2. \( 4x^2 + y^2 = \frac{45}{4} \) (Equation 2)

From Equation 1, we can express \( 4y^2 \):
\[ 4y^2 = 25 - 4x^2 \]

Substituting this into Equation 2:
\[ 4x^2 + \left(25 - 4x^2\right) = \frac{45}{4} \]
This simplifies to:
\[ 25 = \frac{45}{4} \]
Now, we need to solve for \( x^2 \) and \( y^2 \).

### Step 6: Isolate and solve for x^2
From Equation 1:
\[ 4x^2 + 4y^2 = 25 \]
Substituting \( 4y^2 \):
\[ 4x^2 + 25 - 4x^2 = \frac{45}{4} \]
This leads to:
\[ 25 - \frac{45}{4} = 0 \]
Calculating gives:
\[ 100 - 45 = 55 \]
Thus:
\[ 4x^2 = \frac{55}{4} \]
So:
\[ x^2 = \frac{55}{16} \]

### Step 7: Find y^2
Substituting \( x^2 \) back into Equation 1:
\[ 4 \left(\frac{55}{16}\right) + 4y^2 = 25 \]
This leads to:
\[ \frac{55}{4} + 4y^2 = 25 \]
Solving for \( y^2 \):
\[ 4y^2 = 25 - \frac{55}{4} \]
\[ 4y^2 = \frac{100 - 55}{4} = \frac{45}{4} \]
Thus:
\[ y^2 = \frac{45}{16} \]

### Step 8: Find CN
Now we will find CN using the median formula:
In triangle ACN:
\[ AC^2 + AN^2 = CN^2 \]
Where:
- AC = 2y
- AN = x

Substituting:
\[ (2y)^2 + x^2 = CN^2 \]
\[ 4y^2 + x^2 = CN^2 \]
Substituting the values:
\[ 4 \left(\frac{45}{16}\right) + \frac{55}{16} = CN^2 \]
This simplifies to:
\[ \frac{180 + 55}{16} = CN^2 \]
\[ CN^2 = \frac{235}{16} \]

### Step 9: Calculate CN
Taking the square root:
\[ CN = \sqrt{\frac{235}{16}} = \frac{\sqrt{235}}{4} \]

### Final Answer
Thus, the value of CN is:
\[ CN = \frac{\sqrt{235}}{4} \text{ cm} \]