If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is
Given : Two circles (say C1 and C2) with common center as O and
Radius of circle C1, r1 = 4 cm
Radius of circle C2, r2 = 5 cm
Also say AC is the chord of circle C2 which is tangent to circle C1 and OB is the radius of circle to the point of contact of tangent AC .
To find : Length of chord AC
Now, Clearly OB ⏊ AC [ As tangent to at any point on the circle is perpendicular to the radius through point of contact]
So OBC is a right-angled triangle
So, it will satisfy Pythagoras theorem [ i.e. (base)2 + (perpendicular)2 = (hypotenuse)2 ]
i.e.
(OB)2 + (BC)2 = (OC)2
As OB and OC are the radii of circle C1 and C2 respectively.
So
(4)2 + (BC)2 = (5)2
16 + (BC)2 = 25
(BC)2 = 25 - 16 = 9
BC = 3 cm
Also
AB = BC [ Perpendicular through the center to a chord in a circle (C2 in this case) bisects the chord]
And
AC = AB + BC
= AB + AB = 2AB = 2(3) = 6 cm