Lesson OVERVIEW of lessons on the distance from a point to a straight line in a coordinate plane

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OVERVIEW of lessons on the distance from a point to a straight line in a coordinate plane


There are four lessons on this topic in this site:
    - The distance from a point to a straight line in a coordinate plane
    - HOW TO calculate the distance from a point to a straight line in a coordinate plane
    - Using formula for the distance from a point to a straight line in a plane to solve word problems
    - Two chords in a circle are congruent if and only if they are equally remoted from the center

Below these lessons are listed with short annotations


The distance from a point to a straight line in a coordinate plane

Theorem

Let the straight line in a coordinate plane  (Figure 1)  is defined in terms of its linear equation

a%2Ax+%2B+b%2Ay+%2B+c+=+0,                       (1)

where  a,  b and  c  are real numbers,  and let  P = P(x%5B0%5D,y%5B0%5D)  is the point in the coordinate plane
with the coordinates  x%5B0%5D,  y%5B0%5D.  Then the distance from the point  P  to the straight line  (1)  is equal to              

d = abs%28a%2Ax%5B0%5D+%2B+b%2Ay%5B0%5D+%2B+c%29%2Fsqrt%28a%5E2+%2B+b%5E2%29.                (2)


Figure.  The straight line  a%2Ax%2Bb%2Ay%2Bc=0,
the point  P(x%5B0%5D,y%5B0%5D)  and the distance d
from the point  P  to the straight line

      Example.  Find the distance from the point  P = (2,5)  to the straight line   x+-+2y+-2+=+0


HOW TO calculate the distance from a point to a straight line in a coordinate plane

      Problem 1.  Find the distance from the point  P = (8,-1)  to the straight line   2x+-+y+-+2+=+0.

      Problem 2.  Find the distance from the point  P = (-8,-1)  to the straight line   3x+%2B+2y+=+0.

      Problem 3.  Find the distance from the point  P = (2,5)  to the straight line   y+=+%281%2F2%29x+-+1.

      Problem 4.  Find the distance from the point  P = (6,3)  to the straight line   x+-+2.5+=+0.

      Problem 5.  Find the distance from the point  P = (6,3)  to the straight line   y+%2B+2.5+=+0.

      Problem 6.  Find the distance from the point  P = (4,1)  to the straight line   x+-+2y+-+2+=+0.


Using formula for the distance from a point to a straight line in a plane to solve word problems

      Problem 1.  Find the equation of the tangent to the circle   x%5E2+%2B+y%5E2 = 25   passing through the external point  (7,1).

      Problem 2.  Find the equation of the tangent to the circle   x%5E2+%2B+y%5E2+-+4x+%2B2y+-+31 = 0  passing through the external point  (-1,5).


Two chords in a circle are congruent if and only if they are equally remoted from the center

      Problem 1.  A chord  6 cm long is  2 cm from the center of a circle.
                        How long is a chord that is  1  cm from the center of the same circle?

      Problem 2.  Find the radius of a circle in which a chord  10 inches long is  1 inch from the midpoint of the arc it forms?

      Problem 3.  Show that the lines y = 2x − 5 and −2x + 11y = 25 create chords of equal length when they intersect the circle x2 + y2 = 25.


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