Overview

Triangles are easy to draw and easy to underestimate. On the AMC 8 and early AMC 10, a triangle problem is often not asking for a clever theorem. It is asking whether you can choose the right base, find the matching height, keep track of the perimeter, or notice when three lengths cannot form a triangle at all.

This module covers the triangle facts that show up constantly: area, perimeter, the triangle inequality, and the basic shortcuts for isosceles and equilateral triangles.

Definitions and Key Ideas

A triangle is a polygon with three sides and three angles. If its side lengths are $a,b,c$, then its perimeter is

The most important area formula is

The height must be perpendicular to the base. It does not have to be drawn inside the triangle, and the base does not have to be the side that looks horizontal.

For side lengths, the key condition is the triangle inequality:

Equivalently, if two sides have lengths $a$ and $b$, then the third side $c$ must satisfy

The inequalities are strict. If the longest side equals the sum of the other two, the triangle has flattened into a line segment, so it is not a triangle.

Area: Choosing the Right Base

Any side of a triangle can be used as the base. Once the base is chosen, the height is the perpendicular distance from the opposite vertex to the line containing that base.

For example, if $AB=18$ and the altitude from $C$ to $AB$ is $7$, then

A common contest trick is to include extra side lengths or angle information. Use the information the problem gives you immediately.

Triangle Inequality

Suppose two sides are $6$ and $10$. The third side $x$ must satisfy

so

If $x$ must be an integer, the possible values are

There are $11$ such values.

The intuition is rather simple: the two shorter sides have to be long enough to actually meet above the longest side.

Isosceles and Equilateral Triangles

An isosceles triangle has two equal sides. The angles opposite those equal sides are also equal.

So if $AB=AC$, then

An equilateral triangle has all three sides equal and all three angles equal to $60^\circ$. Dropping an altitude in an equilateral triangle splits it into two congruent $30$-$60$-$90$ triangles.

A useful isosceles shortcut: in an isosceles triangle, the altitude from the vertex angle to the base also bisects the base. For instance, in a triangle with sides $10,10,12$, the altitude to the base splits the base into $6$ and $6$. Then the height is

so the area is

Perimeter and Hidden Equations

Perimeter problems often become one-line equations. If an isosceles triangle has equal sides $x,x$ and base $10$, and its perimeter is $38$, then

so

After solving, it is still worth checking the triangle inequality. Here $14+10>14$ and $14+14>10$, so the triangle is valid.

Worked Example

Two sides of a triangle have lengths $5$ and $8$. What are the possible integer lengths for the third side?

By the triangle inequality,

so

The integer values are

so there are

possible lengths.

More Examples

Example 1: Area Without Extra Work

Triangle $ABC$ has $AB=20$. The perpendicular distance from $C$ to line $AB$ is $9$. Find $[ABC]$.

The area is

We do not need to know where the altitude lands on the line.

Example 2: Isosceles Angle Chase

In triangle $ABC$, suppose $AB=AC$ and $\angle A=44^\circ$. Find $\angle B$.

Since $AB=AC$, the base angles are equal, so $\angle B=\angle C$. Thus

and

Example 3: Checking Whether a Triangle Exists

Can a triangle have side lengths $4,7,12$?

No. The two shorter sides have sum

but $11<12$. The two shorter sides cannot reach each other, so no such triangle exists.

Strategies

• If the problem asks for area, look for a base-height pair before doing anything else.
• If a side length is unknown, write the triangle inequality range.
• If two sides are equal, mark the opposite angles equal.
• If two angles are equal, mark the opposite sides equal.
• In an equilateral triangle, immediately write every angle as $60^\circ$.
• In an isosceles triangle, consider dropping the altitude to the base; it often splits the base in half.

Warnings

• Using a slanted side as a height. Heights must be perpendicular.
• Forgetting that triangle inequalities are strict.
• Checking only one triangle inequality. Usually the only dangerous one is whether the two shorter sides are longer than the longest side, but make sure you have identified the longest side correctly.
• Assuming an isosceles triangle is equilateral. Equal sides give equal base angles, not necessarily $60^\circ$ angles.
• Solving a perimeter equation but forgetting to check that the side lengths can actually form a triangle.

Practice Problems