Altitudes of triangle | PRMO 2017 | Question 17

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Try this beautiful problem from the Pre-RMO, 2017 based on Altitudes of triangle.

Altitude of Triangle – PRMO 2017

Suppose the altitudes of a triangle are 10, 12 and 15, find its semi perimeter.

Altitudes

Triangle

Semi-perimeter

But try the problem first…

Answer: is \(\frac{60}{\sqrt{7}}\)

Source

First hint

\(h_a:h_b:h_c\)=10:12:15

or, a:b:c=\(\frac{1}{10} : \frac{1}{12} : \frac{1}{15}\)=6:5:4

or, (a,b,c)=(6k,5k,4k)

or, 2s=15k

Second Hint

\(\Delta=\sqrt{\frac{15k}{2}(\frac{15k}{2}-6k)(\frac{15k}{2}-5k)(\frac{15k}{2}-4k)}\)

or, \(\Delta=\frac{k^215\sqrt{7}}{4}\)

Final Step

\(h_{10}=10 =\frac{2k^2\sqrt{7}\frac{15}{4}}{6k}\)

or, k=\(\frac{8}{\sqrt{7}}\)

or, s=\(\frac{60}{\sqrt{7}}\)