A straight line is the shortest distance between two points. In a triangle, the sum of two sides of the triangle is greater than the third side. Also, a perpendicular drawn from a point to a line is the shortest distance from the point to the line.
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Inequalities
Some Important Facts
Rules useful in handling problems involving inequalities follow.
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That's correct! Let me explain the concepts you mentioned in a bit more detail.
1. The straight line as the shortest distance between two points:
- This is known as the Euclidean distance, a fundamental concept in geometry. It states that the shortest path between two points in a flat, two-dimensional space is a straight line. To calculate the distance between two points (x1, y1) and (x2, y2) in a Cartesian coordinate system, you can use the Euclidean distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
2. Triangle inequality theorem:
- The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Mathematically, for a triangle with sides a, b, and c, the following condition must hold:
a + b > c
b + c > a
c + a > b
This inequality ensures that it is possible to construct a closed figure and that the sides connect properly.
3. Perpendicular drawn from a point to a line as the shortest distance:
- When you have a point and a line in a two-dimensional space, the perpendicular drawn from the point to the line is the shortest distance between the point and the line. This is called the perpendicular distance. The perpendicular distance is the perpendicular line segment that connects the point to the line and forms a right angle with the line.
To find the perpendicular distance from a point (x, y) to a line represented by the equation ax + by + c = 0, you can use the formula:
Distance = | ax + by + c | / √(a^2 + b^2)
This formula uses the concept of projection and the equation of the line to calculate the shortest distance between the point and the line.
Remember that these concepts are fundamental in geometry and mathematics, and they have a wide range of applications in various fields.