This question tests coordinate geometry distance between two points. The distance formula states that d = √[(x₂-x₁)² + (y₂-y₁)²] for points (x₁, y₁) and (x₂, y₂). For points R(-2, 1) and S(4, 4), we calculate: d = √[(4-(-2))² + (4-1)²] = √[(6)² + (3)²] = √[36 + 9] = √45. The distance between R and S is √45, which corresponds to choice B. Note that √45 can be simplified to 3√5, but this simplified form doesn't appear in the choices. Choice A (9) represents the sum of absolute differences |6| + |3|, which is the Manhattan distance, not the Euclidean distance.

This question tests coordinate geometry distance between two points. The distance formula states that for points $(x_1, y_1)$ and $(x_2, y_2)$, the distance is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For $T(4,0)$ and $U(-2,-3)$, we substitute: $d = \sqrt{(-2-4)^2 + (-3-0)^2} = \sqrt{(-6)^2 + (-3)^2}$. This gives us $d = \sqrt{36 + 9} = \sqrt{45}$. The distance is $\sqrt{45}$, which can also be simplified to $3\sqrt{5}$ but this form is not among the choices. The Manhattan distance would be $|4-(-2)| + |0-(-3)| = 6 + 3 = 9$, which appears as choice B but is incorrect for Euclidean distance.

This question tests coordinate geometry distance between two points. The distance formula states that for points $(x_1, y_1)$ and $(x_2, y_2)$, the distance is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For $P(-2,3)$ and $Q(4,-1)$, we substitute: $d = \sqrt{(4-(-2))^2 + (-1-3)^2} = \sqrt{(4+2)^2 + (-4)^2}$. This gives us $d = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52}$. The correct answer is $\sqrt{52}$. A common error would be to calculate the Manhattan distance as $|4-(-2)| + |-1-3| = 6 + 4 = 10$, but this is not the Euclidean distance.

This question tests the distance formula in coordinate geometry. For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. With $L(-3,4)$ and $M(1,-2)$, we calculate: $d = \sqrt{(1-(-3))^2 + (-2-4)^2} = \sqrt{(1+3)^2 + (-6)^2}$. This gives us $d = \sqrt{4^2 + 36} = \sqrt{16 + 36} = \sqrt{52}$. The correct answer is $\sqrt{52}$. The Manhattan distance would be $|1-(-3)| + |-2-4| = 4 + 6 = 10$, which appears as choice B but is incorrect for Euclidean distance.

This question tests coordinate geometry distance calculation. The distance formula for points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For $R(-4,7)$ and $S(-4,-2)$, notice both points have the same x-coordinate of -4, forming a vertical line segment. The distance is $d = \sqrt{(-4-(-4))^2 + (-2-7)^2} = \sqrt{0^2 + (-9)^2} = \sqrt{81} = 9$. Since the points lie on a vertical line, the distance equals the absolute difference of y-coordinates: $|7-(-2)| = |7+2| = 9$. Choice B (81) represents the square of the distance before taking the square root, while choice D (13) might result from incorrectly adding 4 + 9.

This question tests the distance formula in coordinate geometry. The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For $M(0,-3)$ and $N(6,-3)$, notice that both points have the same y-coordinate of -3, making this a horizontal line segment. The distance is $d = \sqrt{(6-0)^2 + (-3-(-3))^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6$. Since the points lie on a horizontal line, the distance is simply the difference in x-coordinates: $|6-0| = 6$. Choice A (36) represents the square of the distance, while choice B ($\sqrt{36}$) is another way to write 6.

This question tests the concept of distance in coordinate geometry. The distance between two points (x1, y1) and (x2, y2) is given by the formula √[(x2 - x1)² + (y2 - y1)²]. For points A(2,5) and B(8,1), the difference in x-coordinates is 8 - 2 = 6, and the difference in y-coordinates is 1 - 5 = -4. Squaring these gives 36 and 16, and adding them results in 52. Taking the square root yields √52, which matches choice A. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |6| + |4| = 10, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

This question tests the concept of distance in coordinate geometry. The distance between two points (x1, y1) and (x2, y2) is given by the formula √[(x2 - x1)² + (y2 - y1)²]. For points C(3,1) and D(-1,4), the difference in x-coordinates is -1 - 3 = -4, and the difference in y-coordinates is 4 - 1 = 3. Squaring these gives 16 and 9, and adding them results in 25. Taking the square root yields √25, which matches choice A. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |4| + |3| = 7, appearing as choice D, but this measures grid path distance, not the straight-line Euclidean distance required here.

This question tests the concept of distance in coordinate geometry. The distance between two points (x1, y1) and (x2, y2) is given by the formula √[(x2 - x1)² + (y2 - y1)²]. For points P(1,1) and Q(4,5), the difference in x-coordinates is 4 - 1 = 3, and the difference in y-coordinates is 5 - 1 = 4. Squaring these gives 9 and 16, and adding them results in 25. Taking the square root yields √25 = 5, which matches choice E. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |3| + |4| = 7, appearing as choice D, but this measures grid path distance, not the straight-line Euclidean distance required here.

This question tests the concept of distance in coordinate geometry. The distance between two points (x1, y1) and (x2, y2) is given by the formula √[(x2 - x1)² + (y2 - y1)²]. For points W(-1,2) and X(3,-1), the difference in x-coordinates is 3 - (-1) = 4, and the difference in y-coordinates is -1 - 2 = -3. Squaring these gives 16 and 9, and adding them results in 25. Taking the square root yields √25 = 5, which matches choice C. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |4| + |3| = 7, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.