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# PLS Help

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1. Points \$X\$ and \$Y\$ lie on a circle centered at \$O,\$ and arc \$XY\$ is \$80^\circ.\$ The circle passing through points \$O,\$ \$X,\$ and \$Y\$ is drawn. Find the measure of arc \$XOY\$ on the smaller circle.

2. Trapezoid \$WXYZ\$ is inscribed in a circle, with \$WX \parallel YZ\$. If arc \$YZ\$ is \$30\$ degrees, arc \$WZ\$ is \$t^2 + 7t\$ degrees, and arc \$XY\$ is \$60 - 4t\$ degrees, find arc \$WTX\$.

3.  In cyclic quadrilateral \$PQRS, (angle P)/2 = (angle Q)/3, = (angle R)/4
Find the largest angle of quadrilateral \$PQRS,\$ in degrees.

4. A regular dodecagon \$P_1 P_2 P_3 \dotsb P_{12}\$ is inscribed in a circle with radius \$1.\$ Compute
\[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\](The sum includes all terms of the form \$(P_i P_j)^2,\$ where \$1 \le i < j \le 12.\$)

Dec 22, 2019

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1. Arc XOY is 220 degrees.

2. Arc WTX is 236 degrees.

Dec 22, 2019
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1. Points \$X\$ and \$Y\$ lie on a circle centered at \$O,\$ and arc \$XY\$ is \$80^\circ.\$ The circle passing through points \$O,\$ \$X,\$ and \$Y\$ is drawn. Find the measure of arc \$XOY\$ on the smaller circle.

2. Trapezoid \$WXYZ\$ is inscribed in a circle, with \$WX \parallel YZ\$. If arc \$YZ\$ is \$30\$ degrees, arc \$WZ\$ is \$t^2 + 7t\$ degrees, and arc \$XY\$ is \$60 - 4t\$ degrees, find arc \$WTX\$.

Dec 23, 2019
edited by Omi67  Dec 23, 2019
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3. Since angle S is equal to (angle Q)/3, by the cycic quadrilateral angle condition, the largest angle is Q = 135.

4. There are 12 digaonals that have a length of P_1 P_2, which from the Sine Law, is sin (15 degrees).  There are 12 diagonals that have a legnth of P_1 P_3, which from the Sine Law, is sin (30 degrees).  We can appy the same reasoning to the other diagonals, which gives us a total sum of

(12 sin 15)^2 + (12 sin 30)^2 + (12 sin 45)^2 + (12 sin 60)^2 + (12 sin 75)^2 + (12 sin 90)^2 + (12 sin 105)^2 + (12 sin 120)^2 + (12 sin 135)^2 + (12 sin 150)^2 + (12 sin 175)^2 = 864.

Since we have double-counted, the answer is 864/2 = 432.

Dec 23, 2019