A Semicircle is inscribed in a right Triangle as shown in figure.

Question

A Semicircle is inscribed in a right Triangle as shown in figure. Calculate the value of  X?

a)  \frac{1}{4}

b)  \frac{2}{7}

c)  \frac{3}{7}

d)  \frac{1}{2}

e)  \frac{4}{7}

chetan2u · Answer

Join Center of the semicircle, D, of say radius r to side BC at E and AC at F. 
As DE and DF are tangents, angles BED and BFD are 90, making BEDF a square with side of length r.

Now take  	riangle CDE  and  	riangle CAB 
They are similar triangles, so  \frac{DE}{AB}=\frac{CE}{CB} 
 =>  \frac{r}{4}=\frac{3-r}{3}…….3r=12-4r…….r=\frac{12}{7}

Once again take  	riangle CDE  to find x
 (3-r)^2+r^2=(r+x)^2………(3-\frac{12}{7})^2+\frac{12}{7}^2=(r+x)^2 
 \frac{9}{7}^2+\frac{12}{7}^2=(r+x)^2 
 \frac{225}{49}=(r+x)^2…….\frac{15}{7}^2=(r+x)^2 
 \frac{15}{7}=r+x=\frac{12}{7}+x 
 x=\frac{3}{7}

C

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A Semicircle is inscribed in a right Triangle as shown in figure.

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A Semicircle is inscribed in a right Triangle as shown in figure.

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