Sum of the squares of the lengths of the tangents from the points `(20,30), (30,40) and (40,50)` to the circle `x ^(2) + y ^(2) = 16` is `100 k, k=`

Answer

To solve the problem, we need to find the sum of the squares of the lengths of the tangents from the points (20,30), (30,40), and (40,50) to the circle given by the equation \(x^2 + y^2 = 16\).

### Step-by-Step Solution:

1. **Identify the Circle's Equation**:
The equation of the circle is given as \(x^2 + y^2 = 16\). This can be rewritten in the standard form as:
\[ S = x^2 + y^2 - 16 = 0 \]

2. **Use the Length of Tangent Formula**:
The length of the tangent from a point \((x_1, y_1)\) to a circle defined by \(S = 0\) is given by:
\[ L = \sqrt{S(x_1, y_1)} \]
where \(S(x_1, y_1) = x_1^2 + y_1^2 - 16\).

3. **Calculate the Length of Tangents**:
- For the point \((20, 30)\):
\[ L_1 = \sqrt{20^2 + 30^2 - 16} = \sqrt{400 + 900 - 16} = \sqrt{1284} \]
- For the point \((30, 40)\):
\[ L_2 = \sqrt{30^2 + 40^2 - 16} = \sqrt{900 + 1600 - 16} = \sqrt{2484} \]
- For the point \((40, 50)\):
\[ L_3 = \sqrt{40^2 + 50^2 - 16} = \sqrt{1600 + 2500 - 16} = \sqrt{4084} \]

4. **Square the Lengths of the Tangents**:
Now we need to find the squares of these lengths:
- \(L_1^2 = 1284\)
- \(L_2^2 = 2484\)
- \(L_3^2 = 4084\)

5. **Sum of the Squares**:
Now we sum these squared lengths:
\[ L_1^2 + L_2^2 + L_3^2 = 1284 + 2484 + 4084 = 7852 \]

6. **Relate to the Given Expression**:
According to the problem, the sum of the squares of the lengths of the tangents is equal to \(100k\):
\[ 7852 = 100k \]
To find \(k\), we divide both sides by 100:
\[ k = \frac{7852}{100} = 78.52 \]

### Final Answer:
Thus, the value of \(k\) is:
\[ \boxed{78.52} \]