Geometry Problems,,,,, 5 in totall

Question

Geometry Problems,,,,, 5 in totall

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698

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1)

In $\triangle ABC$ , we know that $AB = BC = 6\sqrt 3$ and $\angle ABC = 120^\circ.$ Find $AC.$
[asy] pair A,B,C;B = (0,0.5);C = (sqrt(3)/2,0);A = -C;draw(B--C--A--B);label(

2)

In quadrilateral $ABCD,$ $\angle A = \angle C =90^\circ,$ and $BC=CD.$ We know that $AB=\sqrt{13}$ and $AD=\sqrt{5}.$ What is $BC?$
[asy] size(5cm);pair A,B,C,D;A=(0,0);B=(sqrt(13),0);D=(0,sqrt(5));C=(2.8,2.9);draw(A--B--C--D--A);draw(rightanglemark(B,A,D)^^rightanglemark(D,C,B));label(

3)

Two diagonals of quadrilateral $ABCD$ are perpendicular to each other at $O.$ We know $AO=BO,CO=DO,$ and $AB+CD=5\sqrt{2}.$ What is $BD?$
[asy] size(5cm);pair A,B,C,D,O;A=(-1,0);B=(0,1);C=(3,0);D=(0,-3);draw(A--B--C--D--A--C^^B--D);draw(rightanglemark(C,O,B));label(

4)

In $\triangle PQR$ , we have $\angle P = 30^\circ$ , $\angle RQP = 60^\circ$ , and $\angle R=90^\circ$ . Point $X$ is on $\overline{PR}$ such that $\overline{QX}$ bisects $\angle PQR$ . If $PQ = 4\sqrt 3$ , then what is $QX?$

[asy] pair P,Q,R,X;R = (0,0);Q=(0,0.5);P = (sqrt(3)/2,0);X = (0.5/sqrt(3),0);draw(X--Q--P--R--Q);label(

5)

Points $S$ and $T$ are on side $\overline{CD}$ of rectangle $ABCD$ such that $\overline{AS}$ and $\overline{AT}$ trisect $\angle DAB$ . If $CT = 2\sqrt{3}-3$ and $DS = 1$ , then what is the area of $ABCD$ ?