Question

Which of the following points has a distance of 3 from the point (−7, 4) ? Responses (−10, 4) (−10, 4) - no response given (10, 4) (10, 4) - no response given (4, 4) (4, 4) - no response given (−7, −1)

Answer 1

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.

1. For \((-10, 4)\):

\[
9 = (-10 + 7)^2 + (4 - 4)^2 = (-3)^2 + (0)^2 = 9 + 0 = 9 \quad \text{(True)}
\]

2. For \((10, 4)\):

\[
9 = (10 + 7)^2 + (4 - 4)^2 = (17)^2 + (0)^2 = 289 + 0 = 289 \quad \text{(False)}
\]

3. For \((4, 4)\):

\[
9 = (4 + 7)^2 + (4 - 4)^2 = (11)^2 + (0)^2 = 121 + 0 = 121 \quad \text{(False)}
\]

4. 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:

\((-10, 4)\)