The area of an isosceles triangle is 4 square unit. If the length of the third side is 2 unit, the length of each equal side is 

Options

A. 4 units
B. `2sqrt(3)` units
C. `sqrt(17)` units
D. `3sqrt(2)` units

Answer

To find the length of each equal side of the isosceles triangle given that the area is 4 square units and the length of the base (third side) is 2 units, we can follow these steps:

### Step 1: Set up the problem
Let the length of each equal side of the isosceles triangle be \( x \) units. The base of the triangle is given as 2 units.

### Step 2: Use the area formula for the isosceles triangle
The area \( A \) of an isosceles triangle can be calculated using the formula:
\[ A = \frac{1}{4} \sqrt{4a^2 - b^2} \]
where \( a \) is the length of the equal sides and \( b \) is the length of the base.

### Step 3: Substitute the known values into the formula
Given that the area \( A = 4 \) square units and the base \( b = 2 \) units, we can substitute these values into the formula:
\[ 4 = \frac{1}{4} \sqrt{4x^2 - 2^2} \]

### Step 4: Simplify the equation
First, multiply both sides by 4 to eliminate the fraction:
\[ 16 = \sqrt{4x^2 - 4} \]
Next, square both sides to remove the square root:
\[ 256 = 4x^2 - 4 \]

### Step 5: Rearrange the equation
Add 4 to both sides:
\[ 256 + 4 = 4x^2 \]
\[ 260 = 4x^2 \]
Now, divide both sides by 4:
\[ 65 = x^2 \]

### Step 6: Solve for \( x \)
Take the square root of both sides:
\[ x = \sqrt{65} \]

### Conclusion
The length of each equal side of the isosceles triangle is \( \sqrt{65} \) units.

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