Pravoúhlý trojúhelník má obvod 24 cm. Jeho strany jsou v poměru 3 : 4 : 5. Jaký poloměr má kružnice opsaná tomuto trojúhelníku ?

thank you Alan.  I think you have made a small error, the sides of the triangle are in the ratio 3:4:5

but their lengths are 6,8 and 10 cm

so the radius of the circle is 5cm

Google translate from Czech:

A right triangle has a perimeter of 24 cm. Its sides are in the ratio 3: 4: 5. What is the radius of the circle circumscribing the triangle?

First find where the perpendicular bisectors of two of the sides intersect.  This is the centre of the circumscribing circle.  A 3:4:5 circle is a well-known right-angled triangle, so the perpendicular bisectors of the two shorter sides meet at x = 3/2, y = 2  (where the triangle has vertices at (0, 0), (3, 0) and (3, 4)).

The radius is then given by r = √[(3/2 - 0)² + (2 - 0)²]  = 5/2

This answer should be doubled - see Melody's reply.

Google translate to Czech:

První zjistit, kde se kolmé bisectors na dvou stran protínají. To je centrem obalové kružnice.A3: 4: 5 kruh jea dobře známý pravoúhlého trojúhelníku, takže kolmé bisectors na obou kratších stranách setkat při x = 3/2, y = 2 (kdethe trojúhelník má vrcholy na (0, 0), (3, 0) a (3, 4)).

Poloměr je pak dána r = √ [(3/2 - 0) 2 + (2-0) 2] = 5/2

Tato odpověď by měla být dvojnásobná - viz odpověď Melody.

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Here is the pic  

You are right Melody, I forgot about the perimeter, so my answer for the radius should be doubled.

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Very nice, Melody and Alan......I always like this type of problem!!!!