To find out how much space separates two points on a flat surface, we use the idea of distance in math. This concept is crucial in advanced math and physics for things like measuring how fast something moves, the power and direction of electric and gravitational forces, and how signals are handled. Let's examine closely how far apart two spots are from each other.
Introduction to Distance Between Two Points
When we talk about the distance between two points, we're referring to how far apart they are. In geometry, this is found by measuring the length of the straight line that joins them. This method works whether we're dealing with points on a flat surface (like a piece of paper) or in a three-dimensional space.
Distance Between Two Points
The distance between two points is simply the length of the straight line that connects them. This line is special because it's the shortest path between those two points. To find this distance, we just measure how long this straight line is.
Distance Between Two Points Formula
The distance between two points is the length of the straight line that connects them on a flat surface. Imagine two points, A and B, on a graph with coordinates (x1, y1) for A and (x2, y2) for B. To find the distance between A and B, we use this formula:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
This formula helps calculate how far apart any two points are on a graph or coordinate plane.
For distances in a three-dimensional space, we use a similar formula but add the z-coordinates:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2} \]
In a two-dimensional plane, each point is located along horizontal (x) and vertical (y) axes. The x-coordinate measures the distance from the vertical axis (y-axis), while the y-coordinate measures the distance from the horizontal axis (x-axis). Points are written in the form (x, y), where x is the horizontal position and y is the vertical position.
Distance Between Two Points on a Coordinate Plane
In a two-dimensional plane, points are placed along the x-axis and y-axis. The x-coordinate (abscissa) measures how far a point is from the y-axis, while the y-coordinate (ordinate) measures how far a point is from the x-axis. Points on the x-axis have coordinates like (x, 0), and points on the y-axis have coordinates like (0, y).
Important Notes
Point A has coordinates (x1, y1) and point B has coordinates (x2, y2). The formula to find the distance between points A and B is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
The distance of a point (a, b) from:
- The x-axis is |b|.
- The y-axis is |a|.
We use absolute values because distance is always positive.
Solved Examples
Question 1: Find the distance between (1, 2) and (1, 5).
d = √((1 - 1)^2 + (5 - 2)^2) = √(0 + 9) = 3 units
Question 2: Find the distance between (5, 2) and (7, 6).
d = √((7 - 5)^2 + (6 - 2)^2) = √(4 + 16) = √20 = 5 units
Question 3: Show that the points (2, -1), (0, 1), and (2, 3) form a right-angled triangle.
To check if it's a right-angled triangle, we find the distances between each pair of points:
Distance between P(2, -1) and Q(0, 1):
PQ = √((0 - 2)^2 + (1 - (-1))^2) = √(4 + 4) = √8 = 2√2
Distance between Q(0, 1) and R(2, 3):
QR = √((2 - 0)^2 + (3 - 1)^2) = √(4 + 4) = √8 = 2√2
Distance between R(2, 3) and P(2, -1):
RP = √((2 - 2)^2 + (3 - (-1))^2) = √(0 + 16) = 4
Now, apply the Pythagorean theorem:
PQ^2 + QR^2 = PR^2
(2√2)^2 + (2√2)^2 = 4^2
8 + 8 = 16
16 = 16
Therefore, points P, Q, and R satisfy the Pythagorean theorem, confirming that triangle PQR is right-angled.