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.
- is 107
- \(\frac{60}{\sqrt{7}}\)
- is 840
- cannot be determined from the given information
Key Concepts
Altitudes
Triangle
Semi-perimeter
Check the Answer
But try the problem first…
Answer: is \(\frac{60}{\sqrt{7}}\)
PRMO, 2017, Question 17
Geometry Vol I to IV by Hall and Stevens
Try with Hints
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=\(\frac{60}{\sqrt{7}}\)
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA