Question

calcular la distancia que hay entre los puntos n=(2, -1) y k=(9, -9) en el plano cartesiano. escribir la respuesta exacta (y no una aproximación decimal).

Explanation:

Step1: Recall the distance formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the Cartesian plane is given by \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Here, \( N=(2,-1) \) so \( x_1 = 2,y_1=-1 \) and \( K=(9,-9) \) so \( x_2 = 9,y_2=-9 \).

Step2: Substitute the values into the formula

First, calculate \( x_2 - x_1=9 - 2 = 7 \) and \( y_2 - y_1=-9-(-1)=-9 + 1=-8 \).
Then, substitute these into the formula: \( d=\sqrt{(7)^2+(-8)^2} \).

Step3: Simplify the expression inside the square root

Calculate \( 7^2 = 49 \) and \( (-8)^2=64 \). Then \( 49 + 64=113 \). So \( d=\sqrt{113} \).

Answer:

\(\sqrt{113}\)