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

blackpanther 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.

BuiIderBoi Dec 31, 2023