Question 1091564: XZ is a chord which is 12cm long.If the perpendicular distance from the midpoint of the chord to point Y on the circumference of the circle is 4cm,calculate,correct to 1d.p,the perimeter of sector OXYZ.
Answer by KMST(5367)   (Show Source):
You can put this solution on YOUR website! Here is the circle, with the chord XZ, and point Y.
The chord's midpoint, M, and the circle circle center, O, are also labeled.
Here is the perimeter of sector OXYZ, colored in red.
We needed to draw radii OX and OZ, of course.
While we are at it, let's also extend segment YM
to draw full radius OY and full diameter YP,
and draw a few more segments (XY, YZ, and ZP).
We have for radii, and will call their length  cm.
 .
We have four obvious right angles at point M.
We also have right angle YZP, that is not so obvious,
but is a right angle, because it intercepts diameter YP.
That makes for a whole bunch of right triangles,
but if we look look at YZP, YMZ, and ZMP,
we see that they are similar right triangles.
Right triangles ZMP and YMZ are similar,
so their long leg ratios and short leg ratios
(or long leg to short leg ratios) are the same.
Either ratio equation is  ,
 -->  -->  -->  -->  --> 
Now we just need to find the length of arc XYZ,
which by definition is times the measure in radians of
The angle XOZ that contains Y.
We can figure out the measure of the acute angles marked in green in those right triangles:
That corresponds to approximately  radians.
That is the measure of YZX and YPZ, that are inscribed in the circle.
The central angles intercepting the same arcs,
acute angles YOX and YOZ, measure twice as much.
The measure of their sum, the angle XOZ containing point Y, is four times as much,
about  radians.
So, the length, in cm, of arc XYZ is about  .
Now we can calculate the perimeter of sector OXYZ,
by adding the length of arc XYZ,
plus the length of radii OX and OZ :
 .
Rounded to 1 decimal place, it is  .
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