Question

A triangle has sides with lengths $13cm,14cm$ and $15cm$. A circle whose centre lies on the longest side touches the the other two sides. The radius of the circle is (in $cm$) _____.

• A. $\frac{43}{9}$
• B. $\frac{49}{9}$
• C. $\frac{56}{9}$
• D. $\frac{63}{15}$

Answer 1

The correct option is C $\frac{56}{9}$

R.E.F image

draw line from $P$ to $O$. Now

there are $3$ triangles

Area of $△PQR=Ar\left(△POQ\right)+Ar\left(△POR\right)$

Area of $△PQR=\sqrt{s(s-a)(s-b)(s-c)}$

$a=13,b=14,c=15$ on substituting it we get

$S=21\Rightarrow △PQR=\sqrt{21\times 8\times 7\times 6}$

$=\sqrt{3\times 7\times 2^3\times 7\times 2\times 3}$

$=3\times 2^2\times 7=84c{m}^2$

Area of $△lePOQ=\frac{1}{2}\times b\times h$

$=\frac{1}{2}\times 13\times r=\frac{13}{2}rc{m}^2$

Area of $△lePOR=\frac{1}{2}\times b\times r=\frac{1}{2}\times 14\times r=7rc{m}^2$

So, $△PQR=△POQ+△POR$

$84=\frac{13}{2}r+7r$

$84\times 2=14r+13r\Rightarrow 27r=84\times 2$

$n=\frac{84\times 2}{27}$

$r=\frac{56}{9}cm$