PQRS is a cyclic quadrilateral with PQ = 11 RS = 19. M and N are points on PQ and RS respectively such that PM = 6, SN = 7, and MN = 27. The length of the line segment formed when MN is extended from both side until it reaches the circle is ?
Answers
Answer:
Given PQRS is a cyclic quadrilateral.
::Opposite angles of a cyclic quadrilateral are
supplementary
→ Z PSR + < PQR = 180°
Z PQR = 180° - 110°
→ Z PQR = 70°
(2)2 * Z PQR = m(arc PR){The measure of an
inscribed angle is half the measure of the arc
intercepted by it.}
m(arc PR) = 140°
m(arc PQR) = 360° -140° = 220° {Using
Measure of a major arc = 360°- measure of
its corresponding minor arc}
(3)side PQ = side RQ
::m(arc PQ) = m(arc RQ){Corresponding arcs
of congruent chords of a circle (or congruent
circles) are congruent}
= m(arc PQR) = m(arc PQ) + m(arc RQ)
= m(arc PQR) = 2 x m(arc PQ)
= m(arc PQ) = 110°
(4)In A POR,
PQR + < QRP + Z RPQ = 180°{Angle sum
property}
→ Z PRQ + 2 RPQ = 180° - Z PQR
→ 22 PRQ = 180° - 70° {::side PQ = side RQ}
= Z PRQ = 55°
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