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