Question

Which of the following points has a distance of 4 units from the point (-9,5)

Answer 1

To find the points that are a distance of 4 units from the point \((-9, 5)\), we can use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting \((-9, 5)\) as \((x_1, y_1)\) and \(d = 4\):

\[
4 = \sqrt{(x + 9)^2 + (y - 5)^2}
\]

Now, squaring both sides:

\[
16 = (x + 9)^2 + (y - 5)^2
\]

This represents a circle with center at \((-9, 5)\) and a radius of 4. Any point \((x, y)\) that lies on this circle will be at a distance of 4 units from the point \((-9, 5)\).

To find specific points, you can look for integer coordinates that satisfy this equation. Here are examples of points that satisfy this:

1. Move 4 units to the right: \((-9 + 4, 5) = (-5, 5)\)
2. Move 4 units to the left: \((-9 - 4, 5) = (-13, 5)\)
3. Move 4 units up: \((-9, 5 + 4) = (-9, 9)\)
4. Move 4 units down: \((-9, 5 - 4) = (-9, 1)\)
5. Move diagonally (using the Pythagorean theorem):
- Move right 2 units and up 2 units: \((-7, 7)\)
- Move left 2 units and up 2 units: \((-11, 7)\)
- Move right 2 units and down 2 units: \((-7, 3)\)
- Move left 2 units and down 2 units: \((-11, 3)\)

Thus, some example points that are 4 units away from \((-9, 5)\) include:

- \((-5, 5)\)
- \((-13, 5)\)
- \((-9, 9)\)
- \((-9, 1)\)
- \((-7, 7)\)
- \((-11, 7)\)
- \((-7, 3)\)
- \((-11, 3)\)

Any of these points would be correct answers.