Question

which of the following expressions will calculate the length of segment ab if a(-3, 5) and b (4, -6)?

• $sqrt{(-3 + 4)^2 - (5 + 6)^2}$
• none of the choices are correct.
• $sqrt{(-3 - 4)^2 + (5 + 6)^2}$
• $sqrt{(-3 + 4)^2 - (5 - 6)^2}$
• $sqrt{(-3 - 4)^2 + (5 - 6)^2}$

Explanation:

Step1: Recall Distance Formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(A(-3, 5)\) (so \(x_1=-3,y_1 = 5\)) and \(B(4,-6)\) (so \(x_2 = 4,y_2=-6\)).

Step2: Substitute into Formula

Substitute \(x_1=-3,x_2 = 4,y_1 = 5,y_2=-6\) into the formula:
\(x_2 - x_1=4-(-3)=4 + 3=7\) (or \(-3-4=-7\), squaring removes sign), \(y_2 - y_1=-6 - 5=-11\) (or \(5-(-6)=11\), squaring removes sign). So the distance is \(\sqrt{(4 - (-3))^2+(-6 - 5)^2}=\sqrt{(-3 - 4)^2+(5+6)^2}\) (since \((4 - (-3))=-( - 3 - 4)\) and \((-6 - 5)=-(5 + 6)\), squaring makes them equal).
Check the options: The third option is \(\sqrt{(-3 - 4)^2+(5 + 6)^2}\), which matches.

Answer:

\(\boldsymbol{\sqrt{(-3 - 4)^2+(5 + 6)^2}}\) (the third option: \(\sqrt{(-3 - 4)^2+(5 + 6)^2}\))