To find which point has a distance of 3 from the point \((-7, 4)\), we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) is the point \((-7, 4)\) and \((x_2, y_2)\) is the point we are checking.
We need to set the distance \(d\) equal to 3:
\[ 3 = \sqrt{(x_2 + 7)^2 + (y_2 - 4)^2} \]
Squaring both sides gives:
\[ 9 = (x_2 + 7)^2 + (y_2 - 4)^2 \]
Now we will test each of the points.
- For \((-10, 4)\):
\[ 9 = (-10 + 7)^2 + (4 - 4)^2 = (-3)^2 + (0)^2 = 9 + 0 = 9 \quad \text{(True)} \]
- For \((10, 4)\):
\[ 9 = (10 + 7)^2 + (4 - 4)^2 = (17)^2 + (0)^2 = 289 + 0 = 289 \quad \text{(False)} \]
- For \((4, 4)\):
\[ 9 = (4 + 7)^2 + (4 - 4)^2 = (11)^2 + (0)^2 = 121 + 0 = 121 \quad \text{(False)} \]
- For \((-7, -1)\):
\[ 9 = (-7 + 7)^2 + (-1 - 4)^2 = (0)^2 + (-5)^2 = 0 + 25 = 25 \quad \text{(False)} \]
The only point that satisfies the distance condition is \((-10, 4)\).
Thus, the answer is:
\((-10, 4)\)