Euclidean distance is defined as the distance between two points in Euclidean space. To find the distance between two points, the length of the line segment that connects the two points should be measured.
Solved Questions
Here are some sample questions based on the Euclidean distance formula to help you understand the application of the formula in a better way:
Question 1: Calculate the distance between the points (4, 1) and (3, 0).
Using Euclidean Distance Formula:
⇒ d = √(x2 – x1)2 + (y2 – y1)2
⇒ d = √(3 – 4)2 + (0 – 1)2
⇒ d = √(1 + 1)
⇒ d = √2 = 1.414 unit
Question 2: Show that the points A (0, 0), B (4, 0), and C (2, 2√3) are the vertices of an equilateral triangle.
To prove that these three points form an equilateral triangle, we need to show that the distances between all pairs of points, i.e., AB, BC, and CA, are equal.
Distance between points A and B:
AB = √[(4– 0)2 + (0-0)2]
⇒ AB = √16AB = 4 unit
Distance between points B and C:
BC = √[(2-4)^2 + (2√3-0)^2]
⇒ BC = √[4+12] = √16BC = 4 unit
Distance between points C and A:
CA = √[(0-2)2 + (0-2√3)2]
⇒ CA = √[4 + 12] = √16CA = 4 unit
Here, we can observe that all three distances, AB, BC, and CA, are equal.
Therefore, the given triangle is an Equilateral Triangle
Question 3: Mathematically prove Euclidean distance is a non-negative value.
Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:
d = √(x2 – x1)2 + (y2 – y1)2
We know that squares of real numbers are always non-negative.
⇒(x2 – x1)2 >= 0 and (y2 – y1)2 >= 0
⇒ √(x2 – x1)2 + (y2 – y1)2 >= 0As square root of a non-negative number gives a non-negative number,
Therefore Euclidean distance is a non-negative value. It cannot be a negative number.
Question 4: A triangle has vertices at points A(2, 3), B(5, 7), and C(8, 1). Find the length of the longest side of the triangle.
Given, the points A(2, 3), B(5, 7), and C(8, 1) are the vertices of a triangle.
Distance between points A and B:
AB = √[(5-2)2 + (7-3)2]
⇒ AB = √9+16= √25
AB = 5 unitDistance between points B and C:
BC = √[(8-5)2 + (1-7)2]
⇒ BC = √[9+36] = √45
BC = 6.708 unitDistance between points C and A:
CA = √[(8-2)2 + (1-3)2]
⇒ CA = √[36+4] = √40
CA = 6.325 unitTherefore, the length of the longest side of triangle is 6.708 unit.
Practice Problems
These practice problems on Euclidean distance will help you to test your understanding of the concept:
Problem 1: Calculate the Euclidean distance between points P(1, 8, 3) and Q(6, 6, 8).
Problem 2: A car travels from point A(0, 0) to point B(5, 12). Calculate the distance traveled by the car.
Problem 3: An airplane flies from point P(0, 0, 0) to point Q(100, 200, 300). Calculate the distance traveled by the airplane.
Problem 4: A triangle has vertices at points M(1, 2), N(4, 6), and O(7, 3). Find the perimeter of the triangle.
Problem 5: On a graph with points K(2, 3) and L(5, 7), plot these points and calculate the Euclidean distance between them.
Problem 6: A drone needs to fly from point A(1, 1) to point B(10, 10). Find the shortest path the drone should take to conserve battery.
Problem 7: A robotic arm moves from position J(1, 2, 3) to position K(4, 5, 6). Calculate the total distance traveled by the robotic arm.
