Altitude inequalities
Weekly Problem 8 - 2010
Are you able to find triangles such that these five statements are true?
Are you able to find triangles such that these five statements are true?
Age
14 to 16
Challenge level
Problem
A triangle $T$ has an area of $1$cm$^2$. Let $M$ be the product of the perimeter of $T$ and the sum of the three altitudes of $T$. Which of the following statements is false?
A. There are (or there exist) triangles $T$ for which $M> 1000$
B. $M> 6$ for all triangles $T$
C. There are triangles $T$ for which $M=18$
D. $M> 16$ for all right-angled triangles
E. There are triangles $T$ for which $M< 12$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Let the sides of triangle $T$ have lengths $a$, $b$ and $c$ and the corresponding altitudes have lengths $H_a$, $H_b$ and $H_c$.
By the triangle inequality, we have $a+b > c$, $b+c > a$ and $c+a > b$ and so $a+b+c > 2c$, $a+b+c > 2a$ and $a+b+c > 2b$.
Also, since $T$ has area $1$, we have $\frac{1}{2}aH_a=1$, $\frac{1}{2}bH_b=1$ and $\frac{1}{2}cH_c=1$ and so $aH_a=2$, $bH_b=2$ and $cH_c=2$. $$M=(a+b+c)(H_a+H_b+H_c)=(a+b+c)H_a+(a+b+c)H_b+(a+b+c)H_c$$ $$ > 2aH_a+2bH_b+2cH_c=4+4+4=12$$ so $M> 12$ and hence statement E is false.