Question

exercise
what is the distance between ( a(-5,5) ) and ( b(2,13) )? round to the nearest tenth,
justify your steps.
( d=sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} )
( d=sqrt{(2-(-5))^{2}+(13 - 5)^{2}} )
( d=sqrt{7^{2}+8^{2}} )
( d=sqrt{49 + 64} )
( d=sqrt{113} )
( d = 10.63 )
( d = 10.6 )

Explanation:

Step1: Substitute coordinates into distance formula

Given \(A(-5,5)\) and \(B(2,13)\), substitute \(x_1=-5,y_1 = 5,x_2=2,y_2=13\) into \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so \(d=\sqrt{(2-(-5))^2+(13 - 5)^2}\).

Step2: Simplify the expressions inside the square - root

First, \(2-(-5)=2 + 5=7\) and \(13 - 5 = 8\). Then \(d=\sqrt{7^2+8^2}\). Since \(7^2=49\) and \(8^2=64\), we have \(d=\sqrt{49 + 64}\).

Step3: Calculate the sum and the square - root

\(49+64 = 113\), so \(d=\sqrt{113}\approx10.63\). Rounding \(10.63\) to the nearest tenth gives \(d\approx10.6\).

Answer:

The distance between \(A(-5,5)\) and \(B(2,13)\) is approximately \(10.6\).