PLEASE HELP ME WITH IT. In the figure given below , AB and AC are two chords of a circle of center O and radius r. If AB= 2 AC and the perpendiculars drawn fro
PLEASE HELP ME WITH IT.
In the figure given below , AB and AC are two chords of a circle of center O and radius r. If AB= 2 AC and the perpendiculars drawn from the center on these chords are of lengths 'a' and 'b' respectively. PROVE THAT 4b^2= a^2+3r^2
This question is related to lesson CIRCLES
OK. I did it. Now let's see if I can go through it without
getting too complicated.
I think the key to the whole thing is this fact:
A radius drawn perpendicular to a chord bisects the chord.
That tells us several things:
-- OM bisects AB.
'M' is the midpoint of AB.
AM is half of AB.
-- ON bisects AC.
'N' is the midpoint of AC.
AN is half of AC.
-- Since AC is half of AB,
AN is half of AM.
a = b/2
Now look at the right triangle inside the rectangle.
'r' is the hypotenuse, so
a² + b² = r²
But a = b/2, so (b/2)² + b² = r²
(b/2)² = b²/4 b²/4 + b² = r²
Multiply each side by 4: b² + 4b² = 4r²
- - - - - - - - - - -
0 + 5b² = 4r²
original equation: a² + b² = r²
Subtract the last
two equations: -a² + 4b² = 3r²
Add a² to each side: 4b² = a² + 3r² . <=== ! ! !