# 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?

## 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.