GRE Quiz: Coordinate Geometry Distance

What this quiz covers

This quiz focuses on Coordinate Geometry Distance, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $9$
2. $\sqrt{45}$ (correct answer)
3. $\sqrt{13}$
4. $45$
5. $\sqrt{39}$

Explanation: 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.

Question 2

1. $\sqrt{45}$ (correct answer)
2. $9$
3. $45$
4. $\sqrt{27}$
5. $\sqrt{9}$

Explanation: 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.

Question 3

1. $52$
2. $\sqrt{40}$
3. $10$
4. $\sqrt{52}$ (correct answer)
5. $\sqrt{10}$

Explanation: 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.

Question 4

1. $\sqrt{20}$
2. $10$
3. $52$
4. $\sqrt{52}$ (correct answer)
5. $\sqrt{10}$

Explanation: 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.

Question 5

1. $9$ (correct answer)
2. $81$
3. $\sqrt{9}$
4. $13$
5. $\sqrt{13}$

Explanation: 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.

Question 6

1. $36$
2. $\sqrt{36}$
3. $0$
4. $\sqrt{6}$
5. $6$ (correct answer)

Explanation: 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.

Question 7

1. $\sqrt{52}$ (correct answer)
2. $10$
3. $52$
4. $\sqrt{20}$
5. $\sqrt{8}$

Explanation: 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.

Question 8

1. $\sqrt{25}$ (correct answer)
2. $\sqrt{7}$
3. $25$
4. $7$
5. $\sqrt{17}$

Explanation: 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.

Question 9

1. $\sqrt{7}$
2. $\sqrt{25}$
3. $25$
4. $7$
5. $5$ (correct answer)

Explanation: 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.

Question 10

1. $\sqrt{25}$
2. $7$
3. $5$ (correct answer)
4. $25$
5. $\sqrt{13}$

Explanation: 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.

Question 11

1. $\sqrt{28}$
2. $14$
3. $100$
4. $\sqrt{100}$
5. $10$ (correct answer)

Explanation: 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 J(2,-1) and K(-4,7), the difference in x-coordinates is -4 - 2 = -6, and the difference in y-coordinates is 7 - (-1) = 8. Squaring these gives 36 and 64, and adding them results in 100. Taking the square root yields √100 = 10, which matches choice E. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |6| + |8| = 14, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 12

1. $\sqrt{7}$
2. $7$
3. $25$
4. $\sqrt{25}$
5. $5$ (correct answer)

Explanation: 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 L(4,0) and M(0,3), the difference in x-coordinates is 0 - 4 = -4, and the difference in y-coordinates is 3 - 0 = 3. Squaring these gives 16 and 9, 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 |4| + |3| = 7, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 13

1. $\sqrt{49}$
2. $7$ (correct answer)
3. $49$
4. $1$
5. $\sqrt{7}$

Explanation: 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 E(2,-4) and F(2,3), the difference in x-coordinates is 2 - 2 = 0, and the difference in y-coordinates is 3 - (-4) = 7. Squaring these gives 0 and 49, and adding them results in 49. Taking the square root yields √49 = 7, which matches choice B. In this vertical case, the Manhattan distance also equals 7, but a common error might be using √7 (choice E), perhaps from mistakenly squaring only part of the difference.

Question 14

1. $\sqrt{52}$ (correct answer)
2. $10$
3. $\sqrt{20}$
4. $52$
5. $\sqrt{8}$

Explanation: 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 J(-3,2) and K(1,-4), the difference in x-coordinates is 1 - (-3) = 4, and the difference in y-coordinates is -4 - 2 = -6. Squaring these gives 16 and 36, 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 |4| + |6| = 10, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 15

1. $\sqrt{28}$
2. $14$
3. $100$
4. $\sqrt{100}$
5. $10$ (correct answer)

Explanation: 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 Y(0,0) and Z(6,8), the difference in x-coordinates is 6 - 0 = 6, and the difference in y-coordinates is 8 - 0 = 8. Squaring these gives 36 and 64, and adding them results in 100. Taking the square root yields √100 = 10, which matches choice E. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |6| + |8| = 14, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 16

1. $\sqrt{52}$ (correct answer)
2. $10$
3. $52$
4. $\sqrt{20}$
5. $\sqrt{8}$

Explanation: 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 G(5,-1) and H(-1,-5), the difference in x-coordinates is -1 - 5 = -6, and the difference in y-coordinates is -5 - (-1) = -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.

Question 17

1. $\sqrt{61}$ (correct answer)
2. $11$
3. $61$
4. $\sqrt{25}$
5. $\sqrt{11}$

Explanation: 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 U(-2,-2) and V(3,4), the difference in x-coordinates is 3 - (-2) = 5, and the difference in y-coordinates is 4 - (-2) = 6. Squaring these gives 25 and 36, and adding them results in 61. Taking the square root yields √61, which matches choice A. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |5| + |6| = 11, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 18

1. $\sqrt{41}$ (correct answer)
2. $9$
3. $41$
4. $\sqrt{25}$
5. $\sqrt{13}$

Explanation: 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 E(-3,-5) and F(1,0), the difference in x-coordinates is 1 - (-3) = 4, and the difference in y-coordinates is 0 - (-5) = 5. Squaring these gives 16 and 25, and adding them results in 41. Taking the square root yields √41, which matches choice A. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |4| + |5| = 9, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.

Question 19

1. $\sqrt{52}$ (correct answer)
2. $10$
3. $52$
4. $\sqrt{40}$
5. $\sqrt{6}$

Explanation: 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(-2,3) and Q(4,-1), the difference in x-coordinates is 4 - (-2) = 6, and the difference in y-coordinates is -1 - 3 = -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.

Question 20

1. $\sqrt{45}$ (correct answer)
2. $9$
3. $45$
4. $\sqrt{27}$
5. $\sqrt{9}$

Explanation: 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 G(1,-2) and H(7,1), the difference in x-coordinates is 7 - 1 = 6, and the difference in y-coordinates is 1 - (-2) = 3. Squaring these gives 36 and 9, and adding them results in 45. Taking the square root yields √45, which matches choice A. A tempting incorrect option is the Manhattan distance, which adds the absolute differences |6| + |3| = 9, appearing as choice B, but this measures grid path distance, not the straight-line Euclidean distance required here.