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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

 

 Dec 31, 2023
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Identify key points and angles:

 

Point Q is the center of both semicircles.

 

Point R is the foot of the perpendicular from P to line segment AM.

 

Angle QPR is a right angle since PQ is perpendicular to MN.

 

Apply the Pythagorean Theorem to triangles QRP and QRM:

 

Triangle QRP is right-angled at R. We know QR = 4 cm (radius of smaller semicircle) and RP = 4 cm (PR = PM = radius of smaller semicircle). Therefore, PQ = √(RP^2 + QR^2) = √(4^2 + 4^2) = 4√2 cm.

 

Triangle QRM is also right-angled at R. We know QR = 8 cm (radius of bigger semicircle) and RM = 8 cm (RN = RM = radius of bigger semicircle). Therefore, MQ = √(RM^2 + QR^2) = √(8^2 + 8^2) = 8√2 cm.

 

Find the length of PQ:

Since MP and QN are parallel tangents to both semicircles, PQ = MN = MP + QN.

 

Therefore, MN = PQ = 4√2 cm + 8√2 cm = 12√2 cm.

 

Therefore, the length of the perpendicular line PQ is 12√2 cm.

 Dec 31, 2023

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