Question

The radius of the incentre of a triangle is 4 cm and the segments into which one side is divided by the point of the contact are 6 cm and 8 cm. Find the length of other two sides.

• A. One of the other side of triangle is 13 cm.
• B. One of the other side of triangle is 15 cm.
• C. One of the other side of triangle is 12 cm.
• D. One of the other side of triangle is 7 cm.

Answer 1

Answer: (A), (B)

The correct options are
A One of the other side of triangle is 13 cm.
B One of the other side of triangle is 15 cm.

Let $AR=AQ=x$
Since $OP\perp BC$, $OQ\perp CA$ and $OR\perp AB$,
We have $A\left(\square BPOR\right)=24$ sq. cm.
$A\left(\square PCQO\right)=32$ sq. cm.
$A\left(\square AROQ\right)=4x$ sq. cm.
Hence, total area of $\Delta ABC=(24+32+4x)$ sq. cm.
$=(4x+56)$ sq. cm .......... (1)
Also, area of $\Delta ABC=\sqrt{x(x+14)\left(6\right)\left(8\right)}$ .............. (2)
Squaring (1) and (2) and equating, we get
$48x(x+14)=4\left[4\right(x+14){]}^2$
Solve to get $x=7$
Hence, the other two sides are 13 cm. and 15 cm.