The internal base of an isosceles triangle can be whose area is `60 cm^(2)` and the length of one of the equal sides is 13 cm :

Options

A. 20 cm
B. 10 cm
C. 16 cm
D. data insufficient .

Answer

To solve the problem of finding the base of an isosceles triangle given its area and the length of one of the equal sides, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The area of the isosceles triangle (A) = 60 cm²
- The length of one of the equal sides (l) = 13 cm

### Step 2: Use the formula for the area of a triangle
The area (A) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let the base of the triangle be \( b \) and the height be \( h \). Therefore, we can rewrite the formula as:
\[ 60 = \frac{1}{2} \times b \times h \]
This simplifies to:
\[ b \times h = 120 \quad \text{(Equation 1)} \]

### Step 3: Relate the height to the sides of the triangle
In an isosceles triangle, the height can be found using the Pythagorean theorem. The height divides the base into two equal parts, so each part is \( \frac{b}{2} \). Thus, we can write:
\[ h^2 + \left(\frac{b}{2}\right)^2 = l^2 \]
Substituting the known value of \( l \):
\[ h^2 + \left(\frac{b}{2}\right)^2 = 13^2 \]
This simplifies to:
\[ h^2 + \frac{b^2}{4} = 169 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
From Equation 1, we can express \( h \) in terms of \( b \):
\[ h = \frac{120}{b} \]

Now substitute this expression for \( h \) into Equation 2:
\[ \left(\frac{120}{b}\right)^2 + \frac{b^2}{4} = 169 \]

### Step 5: Simplify and solve for \( b \)
Expanding the equation:
\[ \frac{14400}{b^2} + \frac{b^2}{4} = 169 \]

Multiply through by \( b^2 \) to eliminate the fraction:
\[ 14400 + \frac{b^4}{4} = 169b^2 \]

Multiply the entire equation by 4 to clear the fraction:
\[ 57600 + b^4 = 676b^2 \]

Rearranging gives us a standard polynomial form:
\[ b^4 - 676b^2 + 57600 = 0 \]

Let \( x = b^2 \):
\[ x^2 - 676x + 57600 = 0 \]

### Step 6: Use the quadratic formula
To solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -676, c = 57600 \):
\[ x = \frac{676 \pm \sqrt{(-676)^2 - 4 \cdot 1 \cdot 57600}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 676^2 - 4 \cdot 57600 = 456976 - 230400 = 226576 \]
Now calculate \( \sqrt{226576} = 476 \):
\[ x = \frac{676 \pm 476}{2} \]

Calculating the two possible values:
1. \( x = \frac{1152}{2} = 576 \)
2. \( x = \frac{200}{2} = 100 \)

### Step 7: Find the base \( b \)
Since \( x = b^2 \):
1. \( b^2 = 576 \) gives \( b = 24 \)
2. \( b^2 = 100 \) gives \( b = 10 \)

### Conclusion
The possible lengths of the base of the isosceles triangle are \( 24 \, \text{cm} \) and \( 10 \, \text{cm} \).