Geometry Problem. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Sunday, August 14, 2016

### Geometry Problem 1245: Cyclic or Inscribed Quadrilateral, Circle, Arc, Midpoint, Measurement

Labels:
arc,
circle,
cyclic quadrilateral,
inscribed,
measurement,
midpoint

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https://goo.gl/photos/Raksk92QKNNeK9wC6

ReplyDeleteObserve that OE, OF , OG and OH perpendicular to AB, BC, CD and AD.

Draw diameter FF’

We have ∠ (EOF’)= ∠ (B) … ( both angles supplement to ∠ (EOF) )

∠ (HOG)= ∠ (B)…. ( both angles supplement to∠ (D))

Triangles EOF’ congruent to HOG ( case SAS) => EF’= HG

In right triangle FEF’ we have EF^2+EF’^2= EF^2+HG^2= 36+16= 52= 4. Radius^2

Similarly FG^2+EH^2= 4. Radius^2= 25+x^2

So x^2= 52-25=27 and x=3.sqrt(3)

Notice that GE_|_FH, thus EF^2+GH^2=FG^2+EH^2, wherefrom x^2=27.

ReplyDeleteBest regards