Question

7.
(-4, -1) (4, -2)
\\(\sqrt{(-4 - 4)^2 + (-1 - (-2))^2}\\)
= \\(\sqrt{(-8)^2 + (1)^2}\\)
= \\(\sqrt{64 + 1}\\)
= \\(\sqrt{65}\\)

Explanation:

Step1: Identify the distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, the two points are \((- 4,-1)\) and \((4,-2)\), so \(x_1=-4,y_1 = - 1,x_2 = 4,y_2=-2\).

Step2: Substitute values into the formula

Substitute \(x_1=-4,y_1=-1,x_2 = 4,y_2=-2\) into the distance formula:
\(d=\sqrt{(4-(-4))^{2}+(-2 - (-1))^{2}}=\sqrt{(4 + 4)^{2}+(-2 + 1)^{2}}=\sqrt{8^{2}+(-1)^{2}}=\sqrt{64 + 1}=\sqrt{65}\) (Wait, there is a mistake in the original calculation's first step. The correct \(x_2 - x_1\) should be \(4-(-4)=8\) not \(4 - 4=-8\)? Wait no, the two points are \((-4,-1)\) and \((4,-2)\), so \(x_2 - x_1=4-(-4) = 8\), \(y_2 - y_1=-2-(-1)=-1\). But the original calculation wrote \((-4 - 4)\) which is wrong. Let's correct it.
Correct Step1: Substitute \(x_1=-4,y_1 = - 1,x_2=4,y_2=-2\) into distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
\(d=\sqrt{(4-(-4))^{2}+(-2-(-1))^{2}}=\sqrt{(4 + 4)^{2}+(-2 + 1)^{2}}\)

Step2: Calculate each term

Calculate \((4 + 4)^{2}=8^{2}=64\) and \((-2 + 1)^{2}=(-1)^{2}=1\)

Step3: Sum the terms inside the square root

\(64+1 = 65\)

Step4: Take the square root

\(d=\sqrt{65}\) (But the original problem had a sign error in the first step, but the final result is correct because \((-8)^{2}=64\) same as \(8^{2}=64\))

Answer:

The correct distance between the points \((-4,-1)\) and \((4,-2)\) is \(\sqrt{65}\) (Note: The first step in the original calculation had a sign error in the \(x\)-difference, but since squaring eliminates the sign, the final result is correct. The correct \(x\)-difference is \(4-(-4) = 8\), but \((-8)^{2}=64\) same as \(8^{2}=64\), so the final result \(\sqrt{65}\) is correct.)