Question

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

Explanation:

Step1: Recall the distance formula

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

Step2: Substitute the values into the formula

First, calculate the differences in \( x \) and \( y \) coordinates: \( x_2 - x_1=-4 - 2=-6 \), \( y_2 - y_1=-9 - 3=-12 \). Then, square these differences: \( (-6)^2 = 36 \), \( (-12)^2 = 144 \). Add these squared values: \( 36+144 = 180 \). Finally, take the square root: \( \sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx13.42 \) (if we want a decimal approximation) or we can leave it as \( 6\sqrt{5} \). But let's compute it precisely: \( \sqrt{180} = \sqrt{36\times5}=6\sqrt{5}\approx13.42 \), but if we calculate \( \sqrt{180} \) directly, \( 180 = 36\times5 \), so \( \sqrt{180}=6\sqrt{5}\approx13.42 \), but actually \( 6\sqrt{5}\approx13.42 \), but let's do the calculation step by step. Wait, \( (-6)^2=36 \), \( (-12)^2 = 144 \), sum is \( 36 + 144=180 \), then \( \sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx13.42 \), but if we want an exact value, it's \( 6\sqrt{5} \), if we want a decimal, approximately \( 13.42 \). But let's check the calculation again. \( x_1 = 2 \), \( x_2=-4 \), so \( x_2 - x_1=-4 - 2=-6 \), squared is \( 36 \). \( y_1 = 3 \), \( y_2=-9 \), so \( y_2 - y_1=-9 - 3=-12 \), squared is \( 144 \). Sum is \( 36 + 144 = 180 \). Then \( \sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\approx13.42 \). But maybe we can simplify \( \sqrt{180} \) as \( 6\sqrt{5} \) or as a decimal. Let's compute \( 6\sqrt{5} \): \( \sqrt{5}\approx2.236 \), so \( 6\times2.236 = 13.416\approx13.42 \). But the exact value is \( 6\sqrt{5} \) or \( \sqrt{180} \). Wait, but let's check the problem again. The points are \( A(2,3) \) and \( B(-4,-9) \). So applying the distance formula: \( d=\sqrt{(-4 - 2)^2+(-9 - 3)^2}=\sqrt{(-6)^2+(-12)^2}=\sqrt{36 + 144}=\sqrt{180}=6\sqrt{5}\approx13.42 \).

Answer:

The length of \( AB \) is \( 6\sqrt{5} \) (or approximately \( 13.42 \)). If we want the exact value, it's \( 6\sqrt{5} \), and if we want a decimal approximation, it's approximately \( 13.42 \). But since the problem doesn't specify, we can give the exact value \( 6\sqrt{5} \) or the decimal. Let's compute \( \sqrt{180} \) more accurately: \( \sqrt{180}=13.41640786\approx13.42 \). But the exact form is \( 6\sqrt{5} \).