Mathematics RU

The triangle

Does a triangle exist, the angle sum, area by the height and by Heron's formula, medians, midlines and similarity — endless problems with solutions.

How to solve it

Almost every triangle problem rests on a few facts: the angles add up to $180^\circ$, each side is shorter than the other two together, the area is half a side times the height to it, the medians meet at a point that divides them $2 : 1$, and a midline is half the side it is parallel to.

Step by step

Existence: the longest side must be shorter than the sum of the other two. For a third side $c$ with $a$ and $b$ known: $|a - b| < c < a + b$.
Area: with a side and the height to it, $S = \frac{1}{2}ah$. With three sides, Heron's formula.
A height from the area: $h = \frac{2S}{a}$. That gives the height to any side once the area is known.
The centroid cuts off a third of each median, counting from the side; a midline is half the side.
In similar triangles the sides are proportional, and the areas are in the ratio of the square of the scale factor.

A side of the triangle.

The height to that side.

The semi-perimeter: $p = \frac{a + b + c}{2}$.

Example: sides $7, 15, 20$; $p = 21$, $S = \sqrt{21 \cdot 14 \cdot 6 \cdot 1} = 42$, the height to the side $7$ is $h = \frac{2 \cdot 42}{7} = 12$.

Common mistakes

Checking the triangle inequality for a side other than the longest: for the sides $6, 22, 11$ what matters is $22 > 6 + 11$.
Counting the boundary values of a third side: with sides $6$ and $11$ it is strictly between $5$ and $17$, which gives $11$ integer values.
Forgetting the half in the area formula.
Mixing up which part of a median the centroid cuts off: two thirds from the vertex, one third from the side.