I usually don't do this but I'm really sick and missed class. Any help is appreciated!



1) Trapezoid $EFGH$ is inscribed in a circle, with $EF \parallel GH$. If arc $GH$ is $70$ degrees, arc $EH$ is $x^2 - 2x$ degrees, and arc $FG$ is $56 - 3x$ degrees, where $x > 0,$ find arc $EPF$, in degrees.


[asy] unitsize(2 cm); pair A, B, C, D, E; A = dir(170); B = dir(30); D = dir(130); C = dir(70); E = dir(280); draw(Circle((0,0),1)); draw(A--B--C--D--cycle); label("$E$", A, W); label("$F$", B, dir(0)); label("$G$", C, NE); label("$H$", D, NW); dot("$P$", E, S); [/asy]


2) In cyclic quadrilateral $PQRS,$ \[\frac{\angle P}{2} = \frac{\angle Q}{3} = \frac{\angle R}{4}.\]Find the largest angle of quadrilateral $PQRS,$ in degrees.


3) In the diagram, $\angle U = 30^\circ$, arc $XY$ is $170^\circ$, and arc $VW$ is $110^\circ$. Find arc $WY$, in degrees.


[asy] unitsize(2 cm); pair A, B, C, D, E, F; D = dir(160); B = dir(200); E = dir(0); C = dir(270); A = extension(B,C,D,E); F = extension(B,E,C,D); draw(Circle((0,0),1)); draw(C--A--E); draw(C--D--B--E); label("$U$", A, W); label("$V$", B, SW); label("$W$", C, S); label("$X$", D, NW); label("$Y$", E, dir(0)); //label("$F$", F, dir(60)); [/asy]


4) In rectangle $EFGH$, $EH = 3$ and $EF = 4$. Let $M$ be the midpoint of $\overline{EF}$, and let $X$ be a point such that $MH = MX$ and $\angle MHX = 72^\circ$, as shown below. Find $\angle XGH$, in degrees.


[asy] unitsize(1 cm); pair A, B, C, D, M, X; A = (0,3); B = (4,3); C = (4,0); D = (0,0); M = (A + B)/2; X = rotate(26,M)*(D); draw(A--B--C--D--cycle); draw(D--M--X--cycle); draw(C--X); label("$E$", A, NW); label("$F$", B, NE); label("$G$", C, SE); label("$H$", D, SW); label("$M$", M, N); label("$X$", X, S); [/asy]


5) 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 $1 \le i < j \le 12.$)

 Feb 8, 2020

I also need help with these problems, ASAP!


6) If the centre of the circle passing through the origin is $(3,4)$,then the intercepts cut off by the circle on the x-axis and y-axis respectively are what?


7) Let $ABC$ be an acute angled triangle. The circle $L$ with $BC$ as diameter intersects $AC$ again at $P$ and $Q$, respectively. Determine angle $BAC$ (in degrees) given that the orthocentre of triangle $APQ$ lies on $L$.


8) Given the circle with center at $L$, $MRST$ quadrilateral with vertices on the circle $L$, and a circle $O$ inscribed in the quadrilateral, such that
\[ \overline { RM } =17, \ \overline { MT } =19, \ \overline { TS } = 23 \]
What is the value of $ \overline{RT} \times \overline{MS} $? 


9) If the area between three tangent circles of equal radii is $144-\pi^{4} $ and a circle tangent to all three of these circles  has an area A. What is $ \lceil A\rceil$. 


10) If the circle passing through distinct points $(1,t), (t,1),(t,t)\ \forall\ t\in R$ also passes through a fixed point $(a,b)$, then calculate argument of complex number $a+ib$ in radians.


11) In the fig. given below, $O$ is the center of the circle. If $PA = 12$cm, $PC = 15$cm and $CD = 7$cm. Find length of $AB$.


12) Let circle $A$ be a circle with radius $\sqrt{5}$ centered at $(2,0)$ and circle $B$ be a circle with radius $2$ centered at $(-1,0)$ Let the center of circle $A$ be $A_C$ and the center of circle $B$ be $B_C$. The two circles $A$ and $B$ intersect at points $X$ and $Y$. When the area of quadrilateral $X A_C Y B_C$ is expressed in the form $a \sqrt{b} $ where $b$ is nor divisible by the square of any prime, find $a+b$ 

 Feb 8, 2020

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