Strategy: Geometry and Trig Problem-Solving

The Draw-and-Label Strategy

For every geometry problem without a figure, draw your own. Label all given information. This alone prevents most careless errors.

Strategy: Break Complex Shapes into Simple Ones

The ACT loves composite shapes — an L-shaped region, a circle with a square inscribed, a triangle on top of a rectangle.

Step 1: Identify the simple shapes within the composite. Step 2: Calculate each area/perimeter separately. Step 3: Add or subtract as needed.

Strategy: Use Special Right Triangles

Many ACT geometry problems are designed around these ratios:

Triangle	Sides
30-60-90	$x : x\sqrt{3} : 2x$
45-45-90	$x : x : x\sqrt{2}$

If you see $\sqrt{3}$ or $\sqrt{2}$ in the answer choices, a special right triangle is probably involved.

Strategy: Choosing Between Law of Sines and Law of Cosines

Given	Use
Two angles + one side (AAS or ASA)	Law of Sines
One angle + opposite side + another side (SSA)	Law of Sines (watch for ambiguous case)
Two sides + included angle (SAS)	Law of Cosines
Three sides (SSS)	Law of Cosines

Strategy: Complex Number Division

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator:

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(a + bi)(c - di)}{c^2 + d^2}$

The denominator becomes a real number because $(c + di)(c - di) = c^2 + d^2$.

Strategy: Matrix Multiplication Checklist

Before multiplying matrices:

1. Check dimensions: $A$ is $m \times n$, $B$ is $n \times p$ → result is $m \times p$
2. Each entry of the result = dot product of a row from $A$ and a column from $B$
3. If dimensions don't match ($n$ from $A \neq$ rows of $B$), multiplication is undefined

Time Management for Geometry

Geometry questions tend to cluster in the Q20–Q40 range. If you're running low on time:

• Skip problems requiring Law of Cosines or complex setups — they take the longest
• Prioritize coordinate geometry — plugging into formulas is fast and reliable
• Powers of $i$ questions take 10 seconds if you know the cycle — don't skip those