Question

8.
(-5, a) and (1, -4) are two points on a line plotted on a coordinate grid. the distance formula is applied:
\\(\sqrt{(1 - (-5))^2 + (-4 - a)^2}\\)
= \\(\sqrt{(1 + 5)^2 + (-4 - a)^2}\\)
= \\(\sqrt{(6)^2 + (-6)^2}\\)
= \\(\sqrt{36 + 36}\\)
= \\(\sqrt{72} = 6\sqrt{2}\\)

Explanation:

Step1: Analyze 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 \((- 5,a)\) and \((1,-4)\). So, \(x_1=-5,y_1 = a,x_2 = 1,y_2=-4\).

Step2: Substitute into the distance formula

Substitute the values into the formula: \(d=\sqrt{(1-(-5))^2+(-4 - a)^2}=\sqrt{(1 + 5)^2+(-4 - a)^2}=\sqrt{6^2+(-4 - a)^2}\).

Step3: Observe the graph and find the slope or relationship

From the graph, the line passes through \((-5,a)\) and \((1,-4)\) and also passes through \((-2,0)\) (since it crosses the x - axis at \(x=-2\)). The slope between \((-5,a)\) and \((1,-4)\) should be the same as the slope between \((-5,a)\) and \((-2,0)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). The slope between \((-5,a)\) and \((1,-4)\) is \(\frac{-4 - a}{1-(-5)}=\frac{-4 - a}{6}\), and the slope between \((-5,a)\) and \((-2,0)\) is \(\frac{0 - a}{-2-(-5)}=\frac{-a}{3}\). Since they are the same line, \(\frac{-4 - a}{6}=\frac{-a}{3}\). Cross - multiply: \(3(-4 - a)=6(-a)\), \(-12-3a=-6a\), \(3a = 12\), \(a = 2\). Wait, but in the given calculation, we have \(\sqrt{6^2+(-6)^2}\), which implies that \(-4 - a=-6\), so \(a = 2\). Let's check with \(a = 2\).

Step4: Verify the distance calculation

If \(a = 2\), then the distance is \(\sqrt{(1-(-5))^2+(-4 - 2)^2}=\sqrt{6^2+(-6)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\), which matches the given calculation. So the value of \(a\) is \(2\) (if we were to find \(a\)) or the distance between the two points is \(6\sqrt{2}\).

Answer:

If we are finding the distance between the two points \((-5,a)\) and \((1,-4)\) with \(a = 2\), the distance is \(6\sqrt{2}\). If we are finding \(a\), \(a = 2\).