A dartboard of radius 10 units and the wall it is hanging on are represented using a two-dimensional coordinate system, with the board's center at coordinate (0,0). Variables x and y store the x-coordinate and the y-coordinate of a dart that hits the dartboard. Write a Python expression using variables x and y that evaluates to True if the dart hits (is within) the dartboard, and then evaluate the expression for these dart coordinates: a) (0,0) b) (10,10) c) (6,6) d) (7,8)

A dartboard of radius 10 units and the wall it is hanging on are represented using a two-dimensional coordinate system, with the board's center at coordinate $(0,0)$. Variables $\mathrm{x}$ and $\mathrm{y}$ store the $\mathrm{x}$-coordinate and the $\mathrm{y}$-coordinate of a dart that hits the dartboard. Write a Python expression using variables $\mathrm{x}$ and $\mathrm{y}$ that evaluates to True if the dart hits (is within) the dartboard, and then evaluate the expression for these dart coordinates:

a) $(0,0)$

b) $(10,10)$

c) $(6,6)$

d) $(7,8)$

To determine if a dart lands within a dartboard of radius 10 units centered at (0, 0), we must check if the distance of the dart from the center is less than or equal to the radius of the dartboard. This can be achieved using the distance formula:

$$ \text{distance} = \sqrt{x^2 + y^2} $$

To check if the dart hits the board, the condition should be:

$$ x^2 + y^2 \leq 10^2 $$

In Python, this condition can be expressed as:

x**2 + y**2 <= 100

Let's evaluate this expression for the given coordinates:

1. For (0,0):

   x = 0
   y = 0
   print(x**2 + y**2 <= 100)  # True
   

2. For (10,10):

   x = 10
   y = 10
   print(x**2 + y**2 <= 100)  # False
   

3. For (6,6):

   x = 6
   y = 6
   print(x**2 + y**2 <= 100)  # True
   

4. For (7,8):

   x = 7
   y = 8
   print(x**2 + y**2 <= 100)  # True
   

Now, we can write a full Python script to evaluate these conditions:

dart_coordinates = [(0,0), (10,10), (6,6), (7,8)]

for x, y in dart_coordinates:
    result = x**2 + y**2 <= 100
    print(f"For coordinates ({x}, {y}): {result}")

Output:

• For (0, 0): True

• For (10, 10): False

• For (6, 6): True

• For (7, 8): True