Question

directions: find radical form. use y given: j(4, -4), k(-10, -2) find: jk

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 \(J(4,-4)\) and \(K(-10,-2)\), \(x_1 = 4\), \(y_1=-4\), \(x_2=-10\), \(y_2 = - 2\).

Step2: Calculate differences in coordinates

First, find \(x_2 - x_1=-10 - 4=-14\) and \(y_2 - y_1=-2-(-4)=-2 + 4 = 2\).

Step3: Substitute into distance formula

Substitute these values into the formula: \(JK=\sqrt{(-14)^2+2^2}=\sqrt{196 + 4}=\sqrt{200}\).

Step4: Simplify the radical

Simplify \(\sqrt{200}\). We know that \(200=100\times2\), so \(\sqrt{200}=\sqrt{100\times2}=\sqrt{100}\times\sqrt{2}=10\sqrt{2}\)? Wait, no, wait, let's recalculate step 2 and 3. Wait, \(x_2 - x_1=-10 - 4=-14\), squared is \((-14)^2 = 196\), \(y_2 - y_1=-2-(-4)=2\), squared is \(2^2 = 4\). Then sum is \(196 + 4=200\)? Wait, but the marked answer is \(7\sqrt{5}\), maybe I made a mistake. Wait, no, let's check the points again. Wait, maybe the points are \(J(4, - 4)\) and \(K(-3,-2)\)? No, the given is \(K(-10,-2)\). Wait, maybe I miscalculated. Wait, \((-14)^2=196\), \(2^2 = 4\), sum is 200, \(\sqrt{200}=10\sqrt{2}\approx14.14\), and \(7\sqrt{5}\approx15.65\), \(10\sqrt{2}\approx14.14\), \(12\sqrt{3}\approx20.78\). Wait, maybe the points are different. Wait, maybe the x - coordinate of K is - 3? Let's assume that maybe there is a typo, but according to the given points \(J(4,-4)\) and \(K(-10,-2)\), the distance should be \(\sqrt{(-14)^2+2^2}=\sqrt{196 + 4}=\sqrt{200}=10\sqrt{2}\). But the marked answer is \(7\sqrt{5}\), let's check with \(J(4,-4)\) and \(K(-3,-2)\): \(x_2 - x_1=-3 - 4=-7\), \(y_2 - y_1=-2-(-4)=2\), then distance is \(\sqrt{(-7)^2+2^2}=\sqrt{49 + 4}=\sqrt{53}\), no. Wait, \(J(4,-4)\) and \(K(-1,-2)\): \(x_2 - x_1=-1 - 4=-5\), \(y_2 - y_1=-2-(-4)=2\), distance \(\sqrt{25 + 4}=\sqrt{29}\). Wait, maybe \(J(4,-4)\) and \(K(-3,1)\): \(x_2 - x_1=-7\), \(y_2 - y_1=5\), distance \(\sqrt{49 + 25}=\sqrt{74}\). No. Wait, maybe the y - coordinate of J is 4? \(J(4,4)\), \(K(-10,-2)\): \(x_2 - x_1=-14\), \(y_2 - y_1=-6\), distance \(\sqrt{196+36}=\sqrt{232}=2\sqrt{58}\). No. Wait, maybe the problem is to find the distance between \(J(4,-4)\) and \(K(-3,-2)\)? No, the given is \(K(-10,-2)\). Wait, perhaps the original problem has a different set of points. But according to the given points \(J(4,-4)\) and \(K(-10,-2)\), the distance is calculated as follows:

\(x_1 = 4\), \(y_1=-4\); \(x_2=-10\), \(y_2=-2\)

\(\Delta x=x_2 - x_1=-10 - 4=-14\)

\(\Delta y=y_2 - y_1=-2-(-4)=2\)

Distance \(JK=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(-14)^2+2^2}=\sqrt{196 + 4}=\sqrt{200}=10\sqrt{2}\)

But the marked answer is \(7\sqrt{5}\), let's check \(7\sqrt{5}=\sqrt{49\times5}=\sqrt{245}\). So maybe the points are \(J(4,-4)\) and \(K(-3,1)\): \(\Delta x=-7\), \(\Delta y = 5\), \(\sqrt{49 + 25}=\sqrt{74}\), no. \(J(2,-4)\) and \(K(-5,-2)\): \(\Delta x=-7\), \(\Delta y = 2\), \(\sqrt{49+4}=\sqrt{53}\). No. \(J(1,-4)\) and \(K(-6,-2)\): \(\Delta x=-7\), \(\Delta y = 2\), \(\sqrt{49 + 4}=\sqrt{53}\). Wait, maybe the problem is to find the length of a segment with different points. Alternatively, maybe I made a mistake in the sign. Wait, \((x_1 - x_2)\) is the same as \((x_2 - x_1)\) when squared. So \((4-(-10))=14\), squared is 196, \((-4-(-2))=-2\), squared is 4, sum is 200, square root is \(10\sqrt{2}\). So the correct answer based on the given points \(J(4,-4)\) and \(K(-10,-2)\) is \(10\sqrt{2}\).

Answer:

\(10\sqrt{2}\)