+4609 Find the area of a triangle with side lengths 13, 17, and \(12\sqrt2\) .
Semi-perimeter = 1/2[13+17+12sqrt(2)]
S = 23.485......
Area = Sqrt[23.485 x (23.485 - 17) x (23.485 - 12sqrt(2)) x (23.485 - 13)]
Area = 102 square units.
+118609 Our guest has used HERON'S FORMULA
http://www.mathwarehouse.com/geometry/triangles/area/herons-formula-triangle-area.php
Sloppy maths though, which misses part of the structure of the question.
The semi-perimeter,
\(\displaystyle s=\frac{1}{2}(13+17+12\sqrt{2})=15+6\sqrt{2},\)
so applying Heron's formula, the area will equal
\(\displaystyle \sqrt\{ (15+6\sqrt{2})(6\sqrt{2}-2)(6\sqrt{2}+2)(15-6\sqrt{2})\}\, \\=\text{(difference of two squares twice,)}\\\sqrt{(15^{2}-(6\sqrt{2})^{2})((6\sqrt{2})^{2}-2^{2})}\\=\sqrt{153\times68}=\sqrt{9\times17\times4\times17}=3\times2\times17=102.\)
