2.2 Length of a Line Segment
1. What is the Distance Formula?
The distance formula is used to determine the distance, \(d\), between two points in the Cartesian plane. If the points are \(A(x₁,y₁)\) and \(B(x₂,y₂)\), the distance between them is given by:
\[d = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2 }}\]
2. How to Find the Length of a Line Segment?
The length of a line segment is the distance between its endpoints. So, you can use the distance formula to find the length of a line segment. If the endpoints of the line segment are \(A(x₁,y₁)\) and \(B(x₂,y₂)\), the length of the line segment is:
\[length = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2 }}\]
3. What is “Shortest Distance” and How to Calculate It?
The shortest distance from a point to a line is the perpendicular distance from that point to the line. This can be calculated using the distance (Length of line segment) formula mentioned above.
4. How to Calculate Shortest Distance Between a Point and a Line?
The shortest distance from a point to a line is the perpendicular distance from that point to the line. If the line equation is \(Ax + By + C = 0\) and the point is \((x_0,y_0)\), the distance 'd' can be found using the following formula:
\[d =\frac{{|Ax_0+By_0+C|}}{{\sqrt{{A^2+B^2}}}}\]
Example: Let’s find the shortest distance from the point \(P(3,4)\) to the line \(2x + 3y - 6 = 0\).
Using the formula, we get:
\[d =\frac{{|2×3+3×4-6|}}{{\sqrt{{2^2+3^2}}}}=3.32\]
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