Geometry worksheet for calculating distance between two points using the distance formula.

I will solve this geometry problem by calculating the distance between each pair of points using the distance formula. The distance formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

I'll go through each problem one by one.

Problem 1: (4,3) and (6,5)
$d = \sqrt{(6-4)^2 + (5-3)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Problem 2: (5,2) and (3,4)
$d = \sqrt{(3-5)^2 + (4-2)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Problem 3: (0,3) and (2,7)
$d = \sqrt{(2-0)^2 + (7-3)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Problem 4: (4,0) and (2,6)
$d = \sqrt{(2-4)^2 + (6-0)^2} = \sqrt{(-2)^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}$

Problem 5: (0,0) and (6,2)
$d = \sqrt{(6-0)^2 + (2-0)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}$

Problem 6: (-2,6) and (-4,8)
$d = \sqrt{(-4-(-2))^2 + (8-6)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Problem 7: (1,6) and (3,8)
$d = \sqrt{(3-1)^2 + (8-6)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Problem 8: (-5,-2) and (-3,0)
$d = \sqrt{(-3-(-5))^2 + (0-(-2))^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

Problem 9: (1,6) and (4,0)
$d = \sqrt{(4-1)^2 + (0-6)^2} = \sqrt{3^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$

Problem 10: (6,1) and (-3,6)
$d = \sqrt{(-3-6)^2 + (6-1)^2} = \sqrt{(-9)^2 + 5^2} = \sqrt{81 + 25} = \sqrt{106}$

Problem 11: (-2,4) and (2,-4)
$d = \sqrt{(2-(-2))^2 + (-4-4)^2} = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$

Problem 12: (7,6) and (3,-8)
$d = \sqrt{(3-7)^2 + (-8-6)^2} = \sqrt{(-4)^2 + (-14)^2} = \sqrt{16 + 196} = \sqrt{212} = 2\sqrt{53}$

Here are the solutions for all 12 problems:
1. $2\sqrt{2}$
2. $2\sqrt{2}$
3. $2\sqrt{5}$
4. $2\sqrt{10}$
5. $2\sqrt{10}$
6. $2\sqrt{2}$
7. $2\sqrt{2}$
8. $2\sqrt{2}$
9. $3\sqrt{5}$
10. $\sqrt{106}$
11. $4\sqrt{5}$
12. $2\sqrt{53}$