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

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

Details: Click on the figure below.

## Saturday, March 23, 2019

### Geometry Problem 1428: Intersecting Circles, Tangent Line, Triangle, Square, Area

Labels:
area,
congruence,
geometry problem,
intersecting circles,
square,
tangent,
triangle

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Extend EA to T, CA to P, (P, T on circle O)

ReplyDeleteDraw OG ꓕ AP => OP = AC/2

https://photos.app.goo.gl/z8XEgvg4x6Eq1rML8

ReplyDeletesee sketch for position of points M and N

note that OAQ is isosceles right triangle

Let M and N are the projection of Q and O over AC

Triangle ONA congruent to AMQ ( case ASA)

So ON= AM= ½. AE

Area S= ½. ON.AC= ½ x ½ x AC. AC= ¼ S1

Extend AQ to meet Circle Q at X. Drop a perpendicular from C to OA meeting it at Y.

ReplyDeleteLet OA = AQ= QX = r and let AC = y

Tr.s ACY and ACX are similar since < XAY = < ACX = 90

So y/2r = h/y and h = y^2/2r = S1/2r i.e. S1 = 2rh …(1)

But S = rh/2...(2)

From (1) and (2) S1 = 4S

Sumith Peiris

Moratuwa

Sri Lanka