Triangles

Triangles

The single richest figure in CAT geometry. A handful of results — angle sum, area, Pythagoras, and similarity — cover the great majority of questions.

Essentials

• Angle sum $=180°$; the exterior angle equals the sum of the two remote interior angles.
• Area $=\dfrac12\,b\,h$, or Heron's $\sqrt{s(s-a)(s-b)(s-c)}$ with $s=\dfrac{a+b+c}{2}$.
• Pythagoras $a^2+b^2=c^2$; recognise the triples 3-4-5, 5-12-13, 8-15-17.
• Similarity (AA): equal angles ⇒ proportional sides, and areas scale as the square of the side ratio.
• Centroid divides each median in a $2{:}1$ ratio; inradius $r=\dfrac{\text{Area}}{s}$; circumradius $R=\dfrac{abc}{4\,\text{Area}}$.

A worked example

Find the area of a triangle with sides 13, 14, 15.

Semi-perimeter $s=\dfrac{13+14+15}{2}=21$. Then

$\text{Area}=\sqrt{21\,(21-13)(21-14)(21-15)}=\sqrt{21\cdot8\cdot7\cdot6}=\sqrt{7056}=\mathbf{84}.$

(With area 84 you can also get the inradius $r=84/21=4$ and circumradius $R=\frac{13\cdot14\cdot15}{4\cdot84}=8.125$.)

Common traps

• Triangle inequality. Any side must be less than the sum of the other two — some "triangles" can't exist.
• Similar ≠ congruent. Similar triangles share shape, not size; areas go as the square of the ratio.
• Wrong height. The height must be perpendicular to the chosen base.

Checklist

• Use Heron when you know all three sides
• Spot Pythagorean triples to skip arithmetic
• Apply area ratio = (side ratio)² for similar triangles
• Recall centroid 2:1, $r=\text{Area}/s$, $R=abc/4\text{Area}$