Distance from a point to a line
Explanation
We can clearly see the graph of line BC is shown in red colors and its equation is written in the form y=mx+b. This line passes through two points (blue) B and C. Point A is present on the coordinate plane and its coordinates are (x1, y1). D is the intersection of the perpendicular line segment DA with the line BC. Length of the line segment DA (green) is equal to the distance of point A from the line BC. We can move these three free points A, B and C and observe how the position of line BC( and equation)and the length of AD change.
distance of a point from a line
Explore
In the graph above move the points around to explore different position of the perpendicular line and the line it intersects with
The shortest distance from a point to a line
1. ) What is the shortest distance from a point to a line?[br]Discuss and explain fully.
distance of a point from a line
2.) Move the line to have it with a slope of -1 and a y-intercept of 1
What is the slope of the perpendicular line segmant?
3.) Now move the perpendicular so that A is on the coodinate A(3,5)
The distance is shown BUT how can you calculate this distance on the coordinate plane?[br]Explain and discuss
distance of a point from a line
Explanation
If a point is off of a line then you can find the distance from the line by making sure it is on the perpendicular line through that point to the line. [br]Now lets work a few problems involving that fact and a series of steps that is one way you can achieve this goal
Predict
Talk with your group and see if you can come up with a way to find a method[br]Explain below.
distance from a point to a line
Geometry Distance from a point to a line
[br][br]Find the distance from point [i]A[/i] to the given line. Round to the nearest tenth.[br]1.) A(-2,4) to y=2x-2[br][br][br]Type in just the numeric value ( no spaces or d= , before the value)[br][br]
Find the distance from point [i]A[/i] to the given line. Round to the nearest tenth.[br] [br]2.) A(-6,8) to y=-3x+10[br][br][br][br][br][br]
Find the distance from point [i]A[/i] to the given line. Round to the nearest tenth.[br] [br][img]data:image/png;base64,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[/img][br][br][br][br][br][br]
Find the distance from point [i]A[/i] to the given line. Round to the nearest tenth.[br] [br][br][img]data:image/png;base64,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[/img][br][br][br][br][br]
Find the distance from point [i]A[/i] to the given line. Round to the nearest tenth.[br] [br][br][img]data:image/png;base64,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[/img][br][br][br][br][br]
Lets wrap it up
Lets apply our new found knowledge to a practical type problem
Follow this link to check your understanding on a application problem.
https://www.geogebra.org/m/weszmehb