Weekly Problem 8

6764 /www/nrich/html/content/id/6764/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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 $M&gt; 1000$ B. $M&gt; 6$ for all triangles $T$ C. There are triangles $T$ for which $M=18$ D. $M&gt; 16$ for all right-angled triangles E. There are triangles $T$ for which $M&lt; 12$ If you liked this problem, <a href="http://nrich.maths.org/public/viewer.php?obj_id=2925&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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. </mdoxml> 2 3 0 1 1 0 0 Weekly Problem 8 - 2010 Are all these statements about triangles true... 2D Geometry, Shape and Space Mixed triangles Measures and Mensuration Perimeters Secondary Mapping Document Area and volume LS