The key statement in the proof that diameter AB bisects chord CD is, slope of AB = [tex]m1 =\frac{ -4}{3}[/tex] and slope of CD = [tex]m2 =\frac{ 3}{4}[/tex]
What is slope?
The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line.
[tex]m =\frac{ (y2 - y1)}{(x2 - x1)}[/tex]
where, m is the slope of the line.
What is the relation of slopes of 2 perpendicular lines?
The product of slopes of two perpendicular lines is equal to -1.
According to the question
Circle O has a diameter AB with coordinates
A(5, 19) and B(17,3)
Circle O has a chord CD with coordinates
C(11, 1) and D(20.6, 8.2)
now,
[tex]m =\frac{ (y2 - y1)}{(x2 - x1)}[/tex]
Slope of diameter AB = [tex]m1 =\frac{ (3 - 19)}{(17 - 5)}[/tex]
= [tex]m1 =\frac{ ( - 16)}{(12)}[/tex]
= [tex]m1 =\frac{ -4}{3}[/tex]
Slope of chord CD = [tex]m2 =\frac{ (8.2-1)}{(20.6-11)}[/tex]
= [tex]m2 =\frac{ (7.2)}{(9.6)}[/tex]
= [tex]m2 =\frac{ 3}{4}[/tex]
Product of both slope = m1 * m2
= -1
Thus, both are perpendicular to each other .
i.e according to the theorem given
"If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord."
= diameter AB bisects the chord CD
Hence, The key statement in the proof that diameter AB bisects chord CD is, slope of AB = [tex]m1 =\frac{ -4}{3}[/tex] and slope of CD = [tex]m2 =\frac{ 3}{4}[/tex]
To know more about slope here:
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