To determine which point lies on the circle represented by the equation [tex]\( x^2 + (y - 12)^2 = 25^2 \)[/tex], we need to check whether each given point satisfies the circle's equation.
Let's break it down step-by-step for each point:
### For point [tex]\( (20, -3) \)[/tex]:
1. Substitute [tex]\( (x, y) = (20, -3) \)[/tex] into the equation:
[tex]\[ 20^2 + (-3 - 12)^2 = 25^2 \][/tex]
2. Calculate the values:
[tex]\[ 20^2 = 400 \][/tex]
[tex]\[ -3 - 12 = -15 \][/tex]
[tex]\[ (-15)^2 = 225 \][/tex]
3. Add the results:
[tex]\[ 400 + 225 = 625 \][/tex]
4. Compare with [tex]\( 25^2 \)[/tex]:
[tex]\[ 625 = 625 \][/tex]
Since the left-hand side equals the right-hand side, point [tex]\( (20, -3) \)[/tex] lies on the circle.
### For point [tex]\( (-7, 24) \)[/tex]:
1. Substitute [tex]\( (x, y) = (-7, 24) \)[/tex] into the equation:
[tex]\[ (-7)^2 + (24 - 12)^2 = 25^2 \][/tex]
2. Calculate the values:
[tex]\[ (-7)^2 = 49 \][/tex]
[tex]\[ 24 - 12 = 12 \][/tex]
[tex]\[ 12^2 = 144 \][/tex]
3. Add the results:
[tex]\[ 49 + 144 = 193 \][/tex]
4. Compare with [tex]\( 25^2 \)[/tex]:
[tex]\[ 193 \ne 625 \][/tex]
Since the left-hand side does not equal the right-hand side, point [tex]\( (-7, 24) \)[/tex] does not lie on the circle.
### For point [tex]\( (0, 13) \)[/tex]:
1. Substitute [tex]\( (x, y) = (0, 13) \)[/tex] into the equation:
[tex]\[ 0^2 + (13 - 12)^2 = 25^2 \][/tex]
2. Calculate the values:
[tex]\[ 0^2 = 0 \][/tex]
[tex]\[ 13 - 12 = 1 \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
3. Add the results:
[tex]\[ 0 + 1 = 1 \][/tex]
4. Compare with [tex]\( 25^2 \)[/tex]:
[tex]\[ 1 \ne 625 \][/tex]
Since the left-hand side does not equal the right-hand side, point [tex]\( (0, 13) \)[/tex] does not lie on the circle.
### For point [tex]\( (-25, -13) \)[/tex]:
1. Substitute [tex]\( (x, y) = (-25, -13) \)[/tex] into the equation:
[tex]\[ (-25)^2 + (-13 - 12)^2 = 25^2 \][/tex]
2. Calculate the values:
[tex]\[ (-25)^2 = 625 \][/tex]
[tex]\[ -13 - 12 = -25 \][/tex]
[tex]\[ (-25)^2 = 625 \][/tex]
3. Add the results:
[tex]\[ 625 + 625 = 1250 \][/tex]
4. Compare with [tex]\( 25^2 \)[/tex]:
[tex]\[ 1250 \ne 625 \][/tex]
Since the left-hand side does not equal the right-hand side, point [tex]\( (-25, -13) \)[/tex] does not lie on the circle.
### Conclusion:
The only point that satisfies the circle's equation is [tex]\( (20, -3) \)[/tex]. Therefore, the correct answer is:
A. [tex]\( (20, -3) \)[/tex]
