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

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

This Lesson (OVERVIEW of lessons on the distance from a point to a straight line in a coordinate plane) was created by by ikleyn(52776)

: View Source, Show
About ikleyn:

<H2>OVERVIEW of lessons on the distance from a point to a straight line in a coordinate plane</H2>

There are four lessons on this topic in this site:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/The-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>The distance from a point to a straight line in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/HOW-TO-calculate-the-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>HOW TO calculate the distance from a point to a straight line in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/Using-formula-for-the-distance-from-a-point-to-a-straight-line-in-a-plane-to-solve-concrete-problems.lesson>Using formula for the distance from a point to a straight line in a plane to solve word problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/Two-chords-in-a-circle-are-congruent-if-and-only-if-they-are-equally-remoted-from-the-center.lesson>Two chords in a circle are congruent if and only if they are equally remoted from the center</A>

<H3>Below these lessons are listed with short annotations</H3>

<A HREF=https://www.algebra.com/algebra/homework/Vectors/The-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>The distance from a point to a straight line in a coordinate plane</A>

<BLOCKQUOTE><H3>Theorem</H3><TABLE>
 <TR>
 <TD>Let the straight line in a coordinate plane &nbsp;(Figure 1)&nbsp; is defined in terms of its linear equation

{{{a*x + b*y + c = 0}}}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

where &nbsp;a, &nbsp;b and &nbsp;c &nbsp;are real numbers, &nbsp;and let &nbsp;P = P({{{x[0]}}},{{{y[0]}}})&nbsp; is the point in the coordinate plane 
with the coordinates &nbsp;{{{x[0]}}}, &nbsp;{{{y[0]}}}. &nbsp;Then the distance from the point &nbsp;P&nbsp; to the straight line &nbsp;(1)&nbsp; is equal to&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

{{{d}}} = {{{abs(a*x[0] + b*y[0] + c)/sqrt(a^2 + b^2)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

</TD>
 <TD>
{{{drawing( 252, 250, -2.0, 8.1, -4.0, 6.0,
 grid(1), 
 line (-3.6, -2.8, 8.0, 3.0),
 locate (4.5, 2.8, ax+by+c=0),

circle (2, 5, 0.15),
            locate (2.1, 5.6, P),
            locate (3.1, 3.6, d),
        red(line(2, 5, 4, 1))
)}}}

Figure. &nbsp;The straight line &nbsp;{{{a*x+b*y+c=0}}},
the point &nbsp;P({{{x[0]}}},{{{y[0]}}})&nbsp; and the distance {{{d}}}
from the point &nbsp;P&nbsp; to the straight line
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Example. &nbsp;Find the distance from the point &nbsp;P = (2,5) &nbsp;to the straight line &nbsp;&nbsp;{{{x - 2y -2 = 0}}}

<A HREF=https://www.algebra.com/algebra/homework/word/geometry/HOW-TO-calculate-the-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>HOW TO calculate the distance from a point to a straight line in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the distance from the point &nbsp;P = (8,-1) &nbsp;to the straight line &nbsp;&nbsp;{{{2x - y - 2 = 0}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the distance from the point &nbsp;P = (-8,-1) &nbsp;to the straight line &nbsp;&nbsp;{{{3x + 2y = 0}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Find the distance from the point &nbsp;P = (2,5) &nbsp;to the straight line &nbsp;&nbsp;{{{y = (1/2)x - 1}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Find the distance from the point &nbsp;P = (6,3) &nbsp;to the straight line &nbsp;&nbsp;{{{x - 2.5 = 0}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;Find the distance from the point &nbsp;P = (6,3) &nbsp;to the straight line &nbsp;&nbsp;{{{y + 2.5 = 0}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;Find the distance from the point &nbsp;P = (4,1) &nbsp;to the straight line &nbsp;&nbsp;{{{x - 2y - 2 = 0}}}.

<A HREF=https://www.algebra.com/algebra/homework/Vectors/Using-formula-for-the-distance-from-a-point-to-a-straight-line-in-a-plane-to-solve-concrete-problems.lesson>Using formula for the distance from a point to a straight line in a plane to solve word problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find the equation of the tangent to the circle &nbsp;&nbsp;{{{x^2 + y^2}}} = 25&nbsp;&nbsp; passing through the external point &nbsp;(7,1).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Find the equation of the tangent to the circle &nbsp;&nbsp;{{{x^2 + y^2 - 4x +2y - 31}}} = 0 &nbsp;passing through the external point &nbsp;(-1,5).

<A HREF=https://www.algebra.com/algebra/homework/Vectors/Two-chords-in-a-circle-are-congruent-if-and-only-if-they-are-equally-remoted-from-the-center.lesson>Two chords in a circle are congruent if and only if they are equally remoted from the center</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A chord &nbsp;6 cm long is &nbsp;2 cm from the center of a circle.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How long is a chord that is &nbsp;1 &nbsp;cm from the center of the same circle?

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

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

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.