0
$\begingroup$

This practice engineering board exam which I've tried seems very tricky (question no.2). The inscribed circle of $\triangle ABC$ is tangent to $AB$ at $P$ . If the radius is $21$, $AP=23$ and $PB=27$, find the perimeter of $\triangle ABC$ .I've attempted to use the Pythagorean Theorem and Heron's Formula but the problem is that I've missed the values of the two sides $BC$ and $CA$ .Can you please provide solutions by two methods(geometry and trigonometry)?

enter image description here

$\endgroup$
2
$\begingroup$

Let the circle touch side $BC$ and $CA$ at points $Q$ and $R$ respectively. Let $CQ=CR=m$.

$AR=AP=23$ and $BQ=BP=27$.

$AB=50, BC=27+m, CA=23+m$

Now, apply Heron's Formula and then use $\Delta=rs$.

$\endgroup$
0
$\begingroup$

Another approach:

$\tan(\frac{\alpha}2)=\frac{21}{23}$

$\tan(\frac{\beta}2)=\frac{21}{27}$

$\frac{\alpha}2+\frac{\beta}2+\frac{\gamma}2=90^o$

$\tan(\frac{\alpha}2+\frac{\beta}2)=\tan(90-\frac{\gamma}2)=cotan(\frac{\gamma}2)$

Puting values we get $\tan(\frac{\gamma}2)=\frac6{35}$

Now we use this formula:

$\tan(\frac{\gamma}2)=\frac r{p-c}$

Where r=21, p is half perimeter, and $c=AB=23+27=50$

Plugging these values we get:

$p=\frac r{tan(\frac{\gamma}2)}+c=\frac {21}{\frac 6{35}}+50=\frac {7\cdot 35}{2}+50$

So perimeter P is:

$P=2\times \left(\frac {7\cdot 35}{2}+50\right)=7\cdot 35+100=\boxed {345}$

$\endgroup$
7
  • $\begingroup$ Ohh sorry there's typo error the circle is inscribed and only tangent to line AB of Triangle ABC $\endgroup$ – Denver Feb 4 at 12:57
  • $\begingroup$ Ohh sorry there's typo error the circle is inscribed and only tangent to line AB of Triangle ABC $\endgroup$ – Denver Feb 4 at 12:58
  • $\begingroup$ Can anyone again provide a solution or I just consider SIROUS's answer?? $\endgroup$ – Denver Feb 4 at 14:07
  • $\begingroup$ @Denver, did you want a method for solution or just a number to pass the four option test? $\endgroup$ – sirous Feb 6 at 10:42
  • $\begingroup$ @DR SK MOBINUl HAQUE, Thanks for correcting my error $\endgroup$ – sirous Feb 7 at 12:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.