Answer :
The value or measure of ∠PQR in the given circle where PQ & RQ are tangents from same external point Q is 52°, found by applying theorem in circle C and considering quadrilateral CPQR.
What are tangents?
The word "tangent" refers or means "to touch". The Latin word for the same is "tangere". The line that intersects the circle exactly at one point on its circumference or boundary of circle and never enters the circle's interior is a tangent. A circle can have many or infinite tangents. They are always perpendicular to the radius. A tangent of a circle is straight line that touches or intersects externally the circle at only one point without entering the interior of circle.
Consider the circle C, Given that
∠PCR=128°
QP & RQ are tangents from same point Q, which means
QP= QR {tangents from same external point are equal in length}
CP & CR are radius from the centre C & also perpendicular to the tangents, which means:
CP ⟂ QP
CR ⟂ QR
∠CPQ=90°
∠CRQ=90°
Consider the Quadrilateral CPQR,
∠C + ∠P + ∠Q + ∠R = 360° {angle sum property of quadrilateral}
128 + 90 + ∠Q + 90 = 360
128 + 180 +∠Q = 360°
∠Q = 360° - 180 - 128
∠Q = 360° - 308
∠Q = 52°
Therefore the measure of ∠PQR is 52°.
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