Question

find the length of ab with a(2, 3) and b(-4, -9)

Explanation:

Step1: Recall distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 2,y_1 = 3,x_2=- 4,y_2=-9\).

Step2: Substitute values into formula

First, calculate \(x_2 - x_1=-4 - 2=-6\) and \(y_2 - y_1=-9 - 3=-12\). Then, \((x_2 - x_1)^2=(-6)^2 = 36\) and \((y_2 - y_1)^2=(-12)^2 = 144\).

Step3: Sum and take square root

Sum the squares: \(36 + 144=180\). Then, \(d=\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx13.42\) (but we can also simplify or present as exact value. Wait, wait, let's recalculate: \((-4 - 2)=-6\), squared is 36; \((-9 - 3)=-12\), squared is 144. 36 + 144 = 180? Wait, no, 36+144 is 180? Wait 36+144: 36+144 = 180? Wait 144+36 = 180. Then \(\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx13.42\)? Wait no, wait 6 squared is 36, 12 squared is 144, 36 + 144 is 180? Wait 36+144: 36+100=136, 136+44=180. Yes. But wait, maybe I made a mistake. Wait, \(x_1 = 2,x_2=-4\), so \(x_2 - x_1=-4 - 2=-6\), correct. \(y_1 = 3,y_2=-9\), so \(y_2 - y_1=-9 - 3=-12\), correct. Then squared terms: 36 and 144. Sum is 180. Square root of 180: \(\sqrt{36\times5}=6\sqrt{5}\approx13.42\)? Wait no, 6\(\sqrt{5}\) is approximately 62.236 = 13.416, which is approximately 13.42. But wait, maybe the problem expects an exact value or a simplified radical. Wait, 180 can be simplified as \(6\sqrt{5}\) or as a decimal. Wait, let's check again. Wait, maybe I miscalculated the sum. 36 + 144: 36+144 = 180. Yes. So the distance is \(\sqrt{(-4 - 2)^2+(-9 - 3)^2}=\sqrt{(-6)^2+(-12)^2}=\sqrt{36 + 144}=\sqrt{180}=6\sqrt{5}\approx13.42\). But let's compute 6\(\sqrt{5}\): \(\sqrt{5}\approx2.236\), so 62.236 = 13.416, which is approximately 13.42. But maybe the problem wants the exact value or a whole number? Wait, no, 180 is 36*5, so square root is 6\(\sqrt{5}\). Alternatively, maybe I made a mistake in coordinates. Wait, A(2,3) and B(-4,-9). So the difference in x: -4 - 2 = -6, difference in y: -9 - 3 = -12. Squared: 36 and 144. Sum: 180. Square root: \(\sqrt{180}\). Simplify: \(\sqrt{36\times5}=6\sqrt{5}\). So the length of AB is \(6\sqrt{5}\) or approximately 13.42.

Answer:

The length of \(AB\) is \(6\sqrt{5}\) (or approximately \(13.42\)). If we want the exact value, it is \(6\sqrt{5}\), and if we want a decimal approximation, it is approximately \(13.42\).