SAT Geometry & Trigonometry Quick Reference

1. What You Need to Know

Geometry and trig on the SAT are mostly about spotting the right relationship fast: angle facts, triangle similarity/congruence, circle relationships, coordinate geometry, and a small set of trig ideas (mostly right-triangle trig plus radians/arc length/sector area).

Your job is usually to translate a diagram (or word problem) into equations using a few high-yield theorems:

• Angles (vertical, linear pair, parallel lines)
• Triangles (sum of angles, Pythagorean theorem, special right triangles, similarity)
• Circles (radius/diameter, arc/sector, inscribed angles)
• Coordinate geometry (slope, distance, midpoint)
• Trig (SOHCAHTOA, inverse trig, radians basics)

Critical reminder: SAT diagrams are not necessarily to scale unless stated. Don’t “eyeball” lengths or angles.

Core idea you use constantly

When you can show two triangles are similar, you get:

• equal corresponding angles
• proportional corresponding sides

This is one of the biggest time-savers on SAT geometry.

2. Step-by-Step Breakdown

A) General geometry attack plan (works on most questions)

1. Redraw cleanly (even if it’s already drawn). Mark given lengths/angles.
2. Label unknowns with variables (use one variable when possible).
3. Identify the shape family: triangle, circle, parallel lines, coordinate plane, 3D solid.
4. Write the fastest relationship:
   • triangle: angle sum, Pythagorean, special triangle ratios, area formulas
   • circle: central vs inscribed angles, tangent-radius right angle, arc/sector formulas
   • coordinate: slope, distance, midpoint
5. Look for similarity/congruence triggers:
   • parallel lines create equal angles
   • shared angle in nested triangles
   • right angles in circles (diameter subtends a right angle)
6. Solve algebraically and check units/reasonableness.

B) Similar triangles workflow (high frequency)

1. Prove similarity using one of:
   • AA (two angles)
   • SAS (including angle equal and sides proportional)
   • SSS (all sides proportional)
2. Write a correspondence statement (match vertices carefully).
3. Set up a proportion of corresponding sides.
4. Solve for scale factor, then get requested length/ratio.

Micro-example (setup only):
If triangles are similar with scale factor $k$ (big : small), then:

• each corresponding side satisfies $\frac{\text{big}}{\text{small}} = k$
• areas satisfy $\frac{A_{\text{big}}}{A_{\text{small}}} = k^2$

C) Right-triangle trig workflow

1. Confirm it’s a right triangle.
2. Choose the trig ratio based on given/needed sides:
   • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
   • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
   • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
3. Plug in and solve. Use special triangles when possible (faster than decimals).
4. If angle is asked, use inverse trig:
   • $\theta = \sin^{-1}(x)$, $\cos^{-1}(x)$, or $\tan^{-1}(x)$

D) Circle arc/sector workflow

1. Convert degrees to radians if using radian formulas.
2. Use:
   • arc length: $s = r\theta$ (with $\theta$ in radians)
   • sector area: $A = \frac{1}{2}r^2\theta$ (with $\theta$ in radians)
3. If working in degrees, use fraction-of-circle:
   • $s = \frac{\theta}{360^\circ}\cdot 2\pi r$
   • $A = \frac{\theta}{360^\circ}\cdot \pi r^2$

3. Key Formulas, Rules & Facts

A) Angles and lines

B) Triangles (core)

C) Special right triangles (must memorize)

D) Similarity and congruence

E) Quadrilaterals and polygons

F) Circles

G) Coordinate geometry

H) Trigonometry essentials

4. Examples & Applications

Example 1: Similar triangles from parallel lines

A line parallel to one side of a triangle creates a smaller similar triangle. Suppose in triangle $ABC$, a segment $DE$ is drawn with $D$ on $AB$ and $E$ on $AC$, and $DE \parallel BC$. If $AD=3$, $AB=12$, and $AE=4$, find $AC$.

Key insight: Similar triangles: $\triangle ADE \sim \triangle ABC$.

Set up proportion:
$\frac{AD}{AB}=\frac{AE}{AC}$
Substitute:
$\frac{3}{12}=\frac{4}{AC}$
Solve:
$\frac{1}{4}=\frac{4}{AC}$
$AC=16$

Example 2: Coordinate geometry distance

Find the distance between $(-1,2)$ and $(5,-6)$.

Use distance formula:
$d=\sqrt{(5-(-1))^2+(-6-2)^2}$
$d=\sqrt{6^2+(-8)^2}$
$d=\sqrt{36+64}$
$d=\sqrt{100}$
$d=10$

SAT pattern: Many coordinate problems hide a right triangle; distance is just Pythagorean.

Example 3: Right-triangle trig (height)

From a point on the ground, the angle of elevation to the top of a building is $30^\circ$. The point is $40$ units from the base. Find the building height.

Key insight: Opposite is height, adjacent is $40$.

Use tangent:
$\tan(30^\circ)=\frac{h}{40}$
Use special triangle value:
$\tan(30^\circ)=\frac{1}{\sqrt{3}}$
So:
$\frac{1}{\sqrt{3}}=\frac{h}{40}$
$h=\frac{40}{\sqrt{3}}$
Rationalized (if desired):
$h=\frac{40\sqrt{3}}{3}$

Example 4: Arc length and sector area

A circle has radius $6$. A sector has central angle $150^\circ$. Find arc length.

Use degree arc formula:
$s=\frac{150^\circ}{360^\circ}\cdot 2\pi\cdot 6$
Simplify:
$s=\frac{5}{12}\cdot 12\pi$
$s=5\pi$

Variation: If they give $\theta$ in radians, go straight to $s=r\theta$.

5. Common Mistakes & Traps

1. Trusting the picture (not to scale).

   • What goes wrong: You assume a line “looks” perpendicular or a triangle “looks” isosceles.
   • Why wrong: SAT can draw misleading diagrams.
   • Fix: Only conclude special properties if given or proven (right-angle box, tick marks, explicit statements).

2. Mixing up opposite/adjacent in trig.

   • What goes wrong: You pick adjacent relative to the wrong angle.
   • Why wrong: “Adjacent” depends on which acute angle you’re using.
   • Fix: Circle the angle $\theta$, then label opposite/adjacent from that angle.

3. Forgetting the hypotenuse is always opposite the right angle.

   • What goes wrong: Plugging the wrong side into $\sin(\theta)$ or $\cos(\theta)$.
   • Fix: Find the right angle first; the side across from it is the hypotenuse.

4. Using degree formulas with radians (or vice versa).

   • What goes wrong: You use $s=r\theta$ with $\theta$ in degrees.
   • Why wrong: $s=r\theta$ requires radians.
   • Fix: If you see $\pi$ in an angle measure, it’s radians; otherwise convert using $\pi=180^\circ$.

5. Area/volume scaling mistakes in similar figures.

   • What goes wrong: If lengths scale by $k$, you scale area by $k$.
   • Why wrong: area scales by $k^2$, volume by $k^3$.
   • Fix: Ask “Is this a length, area, or volume?” before scaling.

6. Incorrect triangle similarity matching.

   • What goes wrong: You set up proportions with mismatched corresponding sides.
   • Why wrong: Even if triangles are similar, wrong correspondence gives wrong equations.
   • Fix: Write a vertex matching statement (for example, $\triangle ABC \sim \triangle DEF$ means $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$).

7. Slope errors (sign and order).

   • What goes wrong: You compute $\frac{y_1-y_2}{x_2-x_1}$ inconsistently.
   • Why wrong: Swapping only one order flips the sign.
   • Fix: Use the same order on top and bottom: $\frac{y_2-y_1}{x_2-x_1}$.

8. Radius vs diameter confusion in circles.

   • What goes wrong: You use diameter where radius is needed in $\pi r^2$ or $2\pi r$.
   • Fix: If given diameter $d$, convert with $r=\frac{d}{2}$.

6. Memory Aids & Quick Tricks

7. Quick Review Checklist

• You can do angle-chasing with: vertical angles, linear pairs, parallel line angle rules.
• You remember triangle facts: angle sum $180^\circ$, Pythagorean theorem, triangle inequality.
• You have special triangles memorized: $1:1:\sqrt{2}$ and $1:\sqrt{3}:2$.
• You can prove and use similarity (AA is the most common SAT trigger).
• You know circle basics: $C=2\pi r$, $A=\pi r^2$, inscribed angle is half its arc, tangent-radius is $90^\circ$.
• You can use coordinate formulas: slope, distance, midpoint; parallel $m_1=m_2$, perpendicular $m_1m_2=-1$.
• You can use trig ratios correctly relative to the chosen angle.
• You won’t mix radians and degrees: $s=r\theta$ and $A=\frac{1}{2}r^2\theta$ require radians.

You’ve got this: stay systematic, write the relationship first, and let the algebra finish the job.