KCC Quizzes AQQ303 about the radius of a circle

radius is 21.6025718634

Radius will be 21.611 approx, it is difficult to type here so posting the solution in image 

Flip the quarter circle into a semi-circle. This creates two isosceles triangles with base1=42 and base2 equal to 16. Their vertices are at the point O.  All the 4 sides equal the radius R. The difference of the heights is BC or 15. The equation for this difference is:

15 = sqrt(R² – 64) – sqrt(R² – 441)

Solving the above gives R ~ 21.60257

Lets extend BC to intersect horizontal line at E
Therefore, R = sqrt((15+CE)^2 + 8^2)
Also, now R = sqrt(CE^2 + (13+8)^2)

Solving this gives R = 21.60257

The R value is 

Proof:

We applie Pythagore theorem in triangle OAB:

OB²=R²=OA²+AB²=OA²+64 (1)

We applie Pythagore theorem in triangle OED:

OD²=R²=OE²+ED²=OE²+441 (2)

OA=OE+EA=OE+15 (3)

Relations (1) and (3) gives:

R²=(OE+15)²+64=OE²+30.OE+225+64=OE²+30.OE+289 (4)

Relations (2) and (4) gives:

OE²+441= OE²+30.OE+289 so:

30.OE=441-289=152 and OE=152/30 (5)

Relations (5) and (2) gives:

R²=441+(152/30)² gives the R value.

Thank you so much for such an elegant analytical solution MikeLH.  I spent many hours (my wife thinks too many) messing with Pythagoras to come up with an iterative solution (subsequently posted by others) that I didn’t post because I knew intuitively that this should be analytical.  Please award yourself a gold star or have a beer (or both).

Can I ask a favour?  Contact KC and ask him to keep these quizzes coming.  I have used Analog Devices components over many decades but ultimately their funding for this enterprise likely depends on assessed interest and customer uptake.

With kindest regards,

Botha (non-de-plume)

Extending CD till point x on line AO.

Opposite side of rectangle are equal, hence Ax is 15 and Cx is 8. Marking radius r at BO and DO. , making them right angle triangles.

now through pythagorus theorem

r^2 = (15+x) ^2 + AB^2

r^2 = x^2 + 21^ 2

solving both equations, gives r= 21.6

I don't have an independent solution.

Tantalizing explanation, retiredEE.  Would you mind attaching a drawing to help me understand?

When I first looked at this quiz I wondered if there was a way to represent it in such a way as to use defined formulas rather than doing an analysis using the right triangle relationships. By mirroring the image, you can create two isosceles triangles as shown below.

h = height

a = sides = R

b = base = 42 for ΔOD’C’CDO and 16 for ΔOB’BO

We don’t need each height only their difference which equals 15. This gave an f(R) which I submitted to an online equation solver to calculate R.