To find the distance between Point [tex]\( A(2 \sqrt{3}, 5 \sqrt{2}) \)[/tex] and Point [tex]\( B(-2 \sqrt{3}, -\sqrt{2}) \)[/tex], we will use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's break down the calculation step-by-step.
1. Identify the coordinates of the points:
- Point [tex]\(A\)[/tex]: [tex]\((x_1, y_1) = (2 \sqrt{3}, 5 \sqrt{2})\)[/tex]
- Point [tex]\(B\)[/tex]: [tex]\((x_2, y_2) = (-2 \sqrt{3}, -\sqrt{2})\)[/tex]
2. Calculate the differences in each coordinate:
- [tex]\(\Delta x = x_2 - x_1 = -2 \sqrt{3} - 2 \sqrt{3} = -4 \sqrt{3}\)[/tex]
- [tex]\(\Delta y = y_2 - y_1 = -\sqrt{2} - 5 \sqrt{2} = -6 \sqrt{2}\)[/tex]
3. Substitute the differences into the distance formula:
[tex]\[ \text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(-4 \sqrt{3})^2 + (-6 \sqrt{2})^2} \][/tex]
4. Simplify the squares of the differences:
- [tex]\((-4 \sqrt{3})^2 = 16 \cdot 3 = 48\)[/tex]
- [tex]\((-6 \sqrt{2})^2 = 36 \cdot 2 = 72\)[/tex]
5. Add these results:
[tex]\[ 48 + 72 = 120 \][/tex]
6. Take the square root of the sum:
[tex]\[ \text{Distance} = \sqrt{120} \][/tex]
[tex]\[ \text{Distance} \approx 10.9544 \][/tex]
Therefore, the distance from Point [tex]\( A(2 \sqrt{3}, 5 \sqrt{2}) \)[/tex] to Point [tex]\( B(-2 \sqrt{3}, -\sqrt{2}) \)[/tex] is approximately 10.95 units.
Additionally, the differences in the coordinates are:
- [tex]\(\Delta x \approx -6.9282\)[/tex]
- [tex]\(\Delta y \approx -8.4853\)[/tex]
