When I was coaching my Alma-matter’s team for MMC last year. A teacher gave me this paper. I realize to make a solution and publish it here. This could be a help for people doing a self-review.
MTAP has nothing to do with this solution. This is only MY SOLUTION. Anyone who can read this is allowed to make violent reactions/clarifications.
where XY=8, YZ=10, WZ=21.
Label the points of intersections of the circle to the quadrilateral like the figure shown below. Let x=CX and y=CW. XW=x+y.
From the figure, CX=DX, DY=EY, EZ=FZ, FW=CW.
Why? Tangents of the same circle and the same external point are congruent.
Since CX=x, DX=x, DY=8-x.
Similarly, CW=y, FW=y, FZ=21-y
Since DY=EY, EY=8-x
Since FZ=EZ, EZ=21-y
Since XW=x+y, therefore XW=19.
Problem 2: XU and XV are tangents to circle O. XO=17, OU=8
Let’s connect X to O. The line must pass through W as shown in the figure above.
Observe that .º by definition of the tangent line and radius.
From , we can solve for
Observe that WO is also radius of the circle, thus WO=8.
By similar triangle, we can solve for XZ and WZ.
Solving for XZ
Solving for YZ
Solving for XY
An equilateral triangle with sides of length 10 cm is inscribed in a circle. Find the distance from the center of the circle to the side of triangle.
The distance that asked here is the perpendicular or the shortest distance from the center of the circle to the side of the triangle.
In the figure, forms a 30-60-90 triangle. From here, we know that
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