The radii of two concentric circles are 13 cm and 8 cm . AB is a diameter of the bigger circle BD is tangent to the smaller circle touching it at D.Find the length of AD
We draw a line parralel to BD from O on AD at M such that OM || BD
By thales' theorem or basic proportionality theroem , or similarity
AO/BO = AM/DM
But, AO = BO
=> AM/DM = 1 or AM = DM
Also, LMOD = LODM = 900 (alternate angles)
Now, BD2 = OB2 - OD2 = 169 - 64 =105
or, BD = √105 cm
Again, since OM || BD
so, by similarity
OM/BD = AO/AB
=> OM = BD X AO / AB = √105 x 13 / 26 = √105 / 2
or, OM = √105 / 2
So in triangle ODM
DM = √OM2 + OD2 = √105/4 + 64 = √105 + 256 / 2 = √361 / 2
or, DM = √361 / 2
Hence, AD = 2 DM = √361
hope this helps you frnd........
the lenght of ad is 19 cm
in triangle BOD USE THE TANGENT THEORM AND PYTHAGORAS THEOREM TO GET BD = UNDERROOT 105 THEN BE = 2* UNDERROOT 105
THEN IN TRIANGLE ABD USE PYTHAGORAS THEOREM AND GET THE VALUE OF AE THEN TO GET THE VALUE OF AD USE PYTHAGORAS THEOREM IN TRIANGLE ADE AND THE VALUE OF DE IS UNDERROOT 105 THEN THE LENGHT OF AD IS 19
HOPE THIS HELP IF IT HELPS A THUMBS UP !!!
in triangle BOD angle BOD =900 radius divides the drawn to the point of contact of tangent
there fore ,OB // AE this implies that angle BOD=angle AEB=900 and triagles BDO and BEA are similar
AE=2OD by midpoint theorem but OD =8 given there fore AE=2x8=16
in right angle triangle BOD
BD = square root of 105
in triangle ade DE2 +AE2=AD2
AD =square root of 361
Firstly find DB.
Using Pythagoras Theorem, DB^2=OB^2 - OD^2. This gives DB = sqrt(105) = DE.
Without going into the complexity of angles, as in some of the existing answers, I would suggest using the property of Similar triangles. Clearly, AEB and ODB are similar righttriangles. And as, DB~EB, therefore, AE~OD.
DB = sqrt(105), and EB = 2(sqrt(105))
as OD = 8cm, therefore AE = twice(8cm) = 16cm.
Finally, using Pythagoras Theorem in right triangle AED,
AD^2 = AE^2 + ED^2= 256 + 105= 361
Therefore, AD = sqrt(361) = 19cm.