1) In triangle $WXY,$ $WX = 8$ and $XY = 14.$ The circumcircle of triangle $WXY$ is constructed. The tangent to the circumcircle at $X$ is drawn, and the line through $W$ parallel to this tangent intersects $\overline{XY}$ at $Z.$ Find $YZ.$

[asy] unitsize(2 cm); pair A, B, C, D; A = dir(110); B = dir(210); C = dir(330); D = extension(B, C, A, A + rotate(90)*(B)); draw(Circle((0,0),1)); draw(A--B--C--cycle); draw((B + rotate(90)*(B))--(B - rotate(90)*(B))); draw(A--(A + 2.2*(rotate(90)*(B)))); label("$W$", A, N); label("$X$", B, SW); label("$Y$", C, SE); label("$Z$", D, SW); [/asy]

2) Lines $PTQ$ and $PUR$ are tangent to a circle, as shown below.

[asy] unitsize(4 cm); pair A, O, P, Q, R, T, U; P = (0,0); U = (1,0); T = dir(40); O = extension(P, P + dir(20), U, U + (0,1)); A = intersectionpoint((U + 0.1*dir(63))--(U + 5*dir(63)), Circle(O,abs(O - U))); Q = 1.5*T; R = 1.5*U; draw(Q--P--R); draw(Circle(O,abs(O - U))); draw(T--A--U); label("$A$", A, NE); label("$P$", P, SW); label("$Q$", Q, NE); label("$R$", R, E); label("$T$", T, NW); label("$U$", U, S); [/asy]

If $\angle QTA = 41^\circ$ and $\angle RUA = 63^\circ,$ then find $\angle QPR,$ in degrees.

Guest Feb 13, 2020

#1**+1 **

1) By similar triangles, YZ = 1/2*14^2/8 = 49/4.

2) Angle PQR = 90 - (41 + 63)/2 = 38 degrees.

Guest Feb 13, 2020

#2**0 **

Yes, seeing lazily posted nonsense like this on the forum is enough to make me sick too.

Though MaxWong appreciated your coding. Maybe if I was learning coding, which I would like to one day, I would appreciate it too.

But as a maths question, it is close to useless.

Melody Feb 13, 2020

#3**+1 **

I think this is in LaTeX. I can understand the first two lines. But on the second and 4th paragraph it's all giberrish.

AnimalMaster Feb 13, 2020