Try this beautiful problem from Geometry based on the radius of a semi circle and tangent of a circle.
In the right triangle ABC,AC=12,BC=5 and angle C is a right angle . A semicircle is inscribed in the triangle as shown.what is the radius of the semi circle?
Geometry
congruency
similarity
Answer:$\frac{10}{3}$
AMC-8(2017)
Pre College Mathematics
Here O is the center of the semi circle. Join o and D(where D is the point where the circle is tangent to the triangle ) and Join OB.
Can you now finish the problem ..........
Now the $\triangle ODB $and $\triangle OCB$ are congruent
can you finish the problem........
Let x be the radius of the semi circle
Now the $\triangle ODB$ and $\triangle OCB$ we have
OD=OC
OB=OB
$\angle ODB$=$\angle OCB$= 90 degree`
so $\triangle ODB$ and $\triangle OCB$ are congruent (by RHS)
BD=BC=5
And also $\triangle ODA$ and $\triangle BCA$ are similar....
$\frac{8}{12}$=$\frac{x}{5}$
i.e x =$\frac{10}{3}$
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