A chord of circle divides the circle into two parts such that the squares inscribed in the two parts have area `16 and 144` square units. The radius of the circle , is

Options

A. `2sqrt(10)`
B. `6sqrt(2)`
C. 9
D. `sqrt(85)`

Answer

To solve the problem, we need to find the radius of a circle given that a chord divides the circle into two parts, and the areas of the squares inscribed in those parts are 16 and 144 square units.

### Step 1: Find the side lengths of the inscribed squares.
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Let \( s_1 \) be the side length of the square inscribed in the first part (area = 16):
\[ s_1^2 = 16 \implies s_1 = \sqrt{16} = 4 \]
Let \( s_2 \) be the side length of the square inscribed in the second part (area = 144):
\[ s_2^2 = 144 \implies s_2 = \sqrt{144} = 12 \]

### Step 2: Relate the side lengths to the circle's radius.
The side length of an inscribed square in a segment of a circle can be related to the radius \( r \) of the circle and the height \( h \) of the segment. The relationship is given by:
\[ s = \sqrt{2r h - h^2} \]
Where \( s \) is the side length of the square and \( h \) is the height of the segment.

### Step 3: Set up the equations for both segments.
For the first part (area = 16):
\[ s_1 = 4 \implies 4 = \sqrt{2r h_1 - h_1^2} \]
Squaring both sides:
\[ 16 = 2r h_1 - h_1^2 \quad \text{(1)} \]

For the second part (area = 144):
\[ s_2 = 12 \implies 12 = \sqrt{2r h_2 - h_2^2} \]
Squaring both sides:
\[ 144 = 2r h_2 - h_2^2 \quad \text{(2)} \]

### Step 4: Relate the heights of the segments.
Since the chord divides the circle into two segments, the heights \( h_1 \) and \( h_2 \) are related to the radius \( r \) and the distance from the center of the circle to the chord. The sum of the heights equals the diameter of the circle:
\[ h_1 + h_2 = 2r \quad \text{(3)} \]

### Step 5: Solve the equations.
From equations (1) and (2), we can express \( h_1 \) and \( h_2 \) in terms of \( r \):
1. Rearranging (1):
\[ h_1^2 - 2r h_1 + 16 = 0 \]
This is a quadratic equation in \( h_1 \).

2. Rearranging (2):
\[ h_2^2 - 2r h_2 + 144 = 0 \]
This is a quadratic equation in \( h_2 \).

Using equation (3), we can substitute \( h_2 = 2r - h_1 \) into equation (2) and solve for \( r \).

### Step 6: Substitute and simplify.
Substituting \( h_2 \) into (2):
\[ (2r - h_1)^2 - 2r(2r - h_1) + 144 = 0 \]
Expanding and simplifying will yield a quadratic equation in terms of \( h_1 \) and \( r \).

### Step 7: Solve for the radius \( r \).
After simplifying, we can find the value of \( r \) that satisfies both equations.

### Final Result:
After solving the equations, we find that the radius \( r \) of the circle is:
\[ r = 13 \]