Which of the following points has a distance of 3 from the point (−7, 4)?(1 point)
Responses
(−10, 4)
Left Parenthesis negative 10 comma 4 Right Parenthesis
(−7, −1)
Left Parenthesis negative 7 comma negative 1 Right Parenthesis
(4, 4)
Left Parenthesis 4 comma 4 Right Parenthesis
(10, 4)
The distance between two points in a two-dimensional plane can be calculated using the distance formula derived from the Pythagorean theorem:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
Let's calculate the distance from the point \((-7, 4)\) to each of the given points and see which one has a distance of 3.
1. \((-10, 4)\):
\[ d = \sqrt{(-10 - (-7))^2 + (4 - 4)^2} \]
\[ d = \sqrt{(-3)^2 + 0^2} \]
\[ d = \sqrt{9 + 0} \]
\[ d = \sqrt{9} \]
\[ d = 3 \]
2. \((-7, -1)\):
\[ d = \sqrt{(-7 - (-7))^2 + (-1 - 4)^2} \]
\[ d = \sqrt{0^2 + (-5)^2} \]
\[ d = \sqrt{0 + 25} \]
\[ d = \sqrt{25} \]
\[ d = 5 \]
3. \((4, 4)\):
\[ d = \sqrt{(4 - (-7))^2 + (4 - 4)^2} \]
\[ d = \sqrt{(11)^2 + 0^2} \]
\[ d = \sqrt{121 + 0} \]
\[ d = \sqrt{121} \]
\[ d = 11 \]
4. \((10, 4)\):
\[ d = \sqrt{(10 - (-7))^2 + (4 - 4)^2} \]
\[ d = \sqrt{(17)^2 + 0^2} \]
\[ d = \sqrt{289 + 0} \]
\[ d = \sqrt{289} \]
\[ d = 17 \]
Only the distance from \((-7, 4)\) to \((-10, 4)\) is 3. Therefore, the point with a distance of 3 from the point \((-7, 4)\) is \((-10, 4)\).