Apply the Pythagorean Theorem to find the length between (2,7)
and (−2, 0)
. Round your answer to the nearest hundredth, if necessary.(1 point)
Responses
11
11
8.06
8 point 0 6
5.74
5 point 7 4
65
65
To apply the Pythagorean Theorem to find the length of the line segment between the points (2, 7) and (−2, 0), we need to calculate the distance between the two points in the Cartesian coordinate system. This is done by treating the segment as the hypotenuse of a right-angled triangle, with the sides being the differences in the x-coordinates and y-coordinates.
The formula for the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is derived from the Pythagorean Theorem and is given by:
\[ d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2} \]
For our points (2, 7) and (−2, 0), we can plug them into the formula:
\[ d = \sqrt{{(-2 - 2)}^2 + {(0 - 7)}^2} \]
\[ d = \sqrt{{(-4)}^2 + {(-7)}^2} \]
\[ d = \sqrt{16 + 49} \]
\[ d = \sqrt{65} \]
The exact distance between the points is the square root of 65. To round to the nearest hundredth:
\[ \sqrt{65} \approx 8.06 \]
So, the length between the points (2, 7) and (−2, 0) rounded to the nearest hundredth is approximately 8.06.