Length Of The Perpendicular Line Of A Tangent Of Two Semicircles

Find the Length Of The Perpendicular Line Of A Tangent Of Two Semicircles

From the figure, Two semicircles are met at Q. AM = 4 cm is the radius of the smaller semicircle and BN = 8 cm is the radius of the bigger semicircle. MN is the tangent of both semicircles. PQ is perpendicular to tangent MN, then find the length of this perpendicular line
Solution
Draw a parallel line from M, Which is parallel to AB

Here, we got two triangles, triangle MPR and triangle MLN
From the figure
∠AMN = ∠MNB = 90° {MN is tangent of both semicircles}
AM = 4 cm {radius of the smaller semicircle}
AM = RQ = BL = 4 cm {AM, PQ & BL parallel lines and AB is parallel to ML}
Now consider triangle MPR and triangle MLN

Triangle MPR and triangle MLN are similar triangles
Because
∠RMP = ∠MLMN {common for both triangles}
∠MPR = ∠MNL = 90° {PQ and NM are parallel lines}
Then
PR/NL = MR/ML
⇒ PR/4 = 4/12
Or, PR/4 = 1/3
So, PR = 4/3 cm
Then
PQ = RQ + PR = 4 + 4/3
PQ = 16/3 cm
So, length of the perpendicular line is 16/3 cm