Model Comparison: Geometry

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Gemini 3 Pro

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What You Need to Know

• Geometry is a major component: Geometry (Plane and Coordinate) makes up roughly 35–40% of the ACT Math section (about 20–24 questions). Unlike the SAT, the ACT typically does not provide a reference sheet with formulas at the beginning of the test. You must memorize them.
• Figures are usually scale-trustworthy: On the ACT, geometric figures are generally drawn to scale unless explicitly marked "Not drawn to scale." You can often use visual estimation to eliminate illogical answer choices.
• Coordinate geometry is frequent: You will need to seamlessly switch between algebraic equations and their geometric representations on the xy-plane.

Angles and Parallel Lines

Understanding the relationships between angles is the foundation of ACT geometry. These rules often serve as the first step in multi-step problems.

Key Relationships

• Vertical Angles: Angles opposite each other when two lines intersect. They are always equal.
• Supplementary Angles: Angles that add up to 180^{\circ} (a straight line).
• Complementary Angles: Angles that add up to 90^{\circ}.
• Angles around a point: Sum to 360^{\circ}.

Parallel Lines and Transversals

When a transversal line cuts across two parallel lines, specific angle pairs are formed:

• Alternate Interior Angles: Equal (Z-pattern).
• Corresponding Angles: Equal (F-pattern).
• Consecutive Interior Angles: Supplementary (add to 180^{\circ}).

Exam Focus

• Why it matters: This is the bedrock of plane geometry. ACT questions often disguise these lines within polygons or trapezoids.
• Typical question patterns: You are given a complex figure with one or two known angles and asked to find a distant unknown angle by "walking" values through the diagram using parallel line theorems.
• Common mistakes: Assuming lines are parallel just because they look parallel. Only apply these rules if arrows indicate parallel lines or the text explicitly states AB \parallel CD.

Properties of Triangles and Pythagorean Theorem

Triangles are the most tested shape on the ACT. Mastery of right triangles is non-negotiable.

General Triangle Properties

• Angle Sum: The sum of interior angles is always 180^{\circ}.
• Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• Triangle Inequality Theorem: The sum of the lengths of any two sides must be greater than the length of the third side (a + b > c).
• Isosceles Triangles: If two sides are equal, the angles opposite them are also equal.

The Pythagorean Theorem

For any right-angled triangle with legs a and b and hypotenuse c:
a^2 + b^2 = c^2

Common Pythagorean Triples: Memorizing these saves time on calculation:

• 3-4-5 (and multiples like 6-8-10)
• 5-12-13
• 8-15-17

Special Right Triangles

These appear frequently and allow you to solve for sides with only one side length given.

• 30-60-90 Triangle:
  • Sides are in the ratio x : x\sqrt{3} : 2x
  • The hypotenuse is twice the shortest leg.
• 45-45-90 Triangle (Isosceles Right):
  • Sides are in the ratio x : x : x\sqrt{2}
  • The hypotenuse is the leg times \sqrt{2}.

Exam Focus

• Why it matters: Nearly every polygon problem can be broken down into triangles.
• Typical question patterns: Finding the height of an isosceles triangle, or finding the distance between two points (which creates a right triangle).
• Common mistakes: Confusing the legs with the hypotenuse in the Pythagorean theorem. Always ensure c is the side opposite the 90^{\circ} angle.

Congruence and Similarity

• Congruence (\cong): Identical shape and size. Corresponding sides and angles are equal.
• Similarity (\sim): Same shape, different size. Corresponding angles are equal, but sides are proportional.

Ratios in Similar Triangles

If two triangles are similar with a side ratio of a:b:

• The ratio of their perimeters is a:b.
• The ratio of their areas is a^2:b^2.

Exam Focus

• Typical question patterns: "Shadow problems" (a pole casts a shadow, a person casts a shadow; find the height of the pole). Also, triangles nested inside other triangles using parallel lines.
• Common mistakes: Forgetting to square the ratio when comparing areas. If side lengths triple (3\times), the area becomes 9\times larger.

Trigonometric Ratios and Right Triangles

For the standard ACT math section, trigonometry is primarily focused on right triangles, though unit circle concepts appear in harder questions.

SOH CAH TOA

For a right triangle with an acute angle \theta:

• \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
• \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
• \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

Reciprocal Identities

Occasionally, the ACT tests the reciprocal functions:

• \csc(\theta) = \frac{1}{\sin(\theta)}
• \sec(\theta) = \frac{1}{\cos(\theta)}
• \cot(\theta) = \frac{1}{\tan(\theta)}

Fundamental Identity

\sin^2(\theta) + \cos^2(\theta) = 1

Exam Focus

• Typical question patterns: You are given two sides of a right triangle and asked for \tan(\theta) , or given a scenario involving a ladder leaning against a wall.
• Common mistakes: Calculator mode errors. Ensure your calculator is in DEGREE mode for geometry problems involving triangles, unless the question specifically specifies radians.

Properties of Circles

Essential Formulas

• Area: A = \pi r^2
• Circumference: C = 2\pi r or C = \pi d
• Diameter: d = 2r

Arcs and Sectors

Instead of memorizing complex formulas, think in fractions of the 360^{\circ} total.

• Arc Length: Fraction of the circumference.
  \text{Arc Length} = \frac{\text{central angle}}{360^{\circ}} \times 2\pi r
• Sector Area: Fraction of the total area (like a slice of pizza).
  \text{Sector Area} = \frac{\text{central angle}}{360^{\circ}} \times \pi r^2

Tangents and Chords

• A radius drawn to a point of tangency is perpendicular (90^{\circ}) to the tangent line.

Exam Focus

• Typical question patterns: Finding the area of a shaded region (usually Area of Square minus Area of Circle, or vice versa).
• Common mistakes: Mixing up diameter and radius. If the problem gives the diameter, immediately divide by 2 to write down the radius.

Surface Area and Volume of Solids

Volume Formulas

• Prisms and Cylinders: V = \text{Area of Base} \times \text{Height}
  • Cylinder specific: V = \pi r^2 h
• Pyramids and Cones: V = \frac{1}{3} \times \text{Area of Base} \times \text{Height}
• Sphere: V = \frac{4}{3}\pi r^3

Surface Area

• Conceptually, this is the sum of the areas of all faces (the "wrapping paper").
• Rectangular Prism: SA = 2(lw + lh + wh)

Exam Focus

• Typical question patterns: Questions often ask how the volume changes if the dimensions change (e.g., "If the radius of a cylinder is doubled, how does the volume change?"). Since r is squared in the formula, the volume quadruples.

Coordinate Geometry

This section links algebra with geometry.

Fundamental Formulas

• Slope (m): m = \frac{y2 - y1}{x2 - x1}
• Midpoint Formula: Average of the coordinates.
  M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2})
• Distance Formula: Derived from Pythagorean theorem.
  d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Lines

• Slope-Intercept Form: y = mx + b
• Parallel Lines: Have equal slopes.
• Perpendicular Lines: Have negative reciprocal slopes (e.g., 2 and -\frac{1}{2}).

Exam Focus

• Typical question patterns: Finding the fourth vertex of a parallelogram, or determining the equation of a line perpendicular to a given line.
• Common mistakes: Forgetting the negative sign when calculating perpendicular slopes.

Equations of Conic Sections

The ACT specifically focuses on Circles and Parabolas, with occasional Ellipse questions.

Circle Equation

Standard form for a circle with center (h, k) and radius r:
(x - h)^2 + (y - k)^2 = r^2

Parabola

• Vertex Form: y = a(x-h)^2 + k , where (h,k) is the vertex.

Ellipse (Rare but possible)

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

• Center is (h,k). a and b relate to the width and height of the ellipse.

Exam Focus

• Typical question patterns: You are given an equation like (x-3)^2 + (y+2)^2 = 16 and asked for the center and radius.
  • Center: (3, -2) (watch the signs!)
  • Radius: 4 (square root of 16)

Quick Review Checklist

Before the exam, verify you can answer these questions:

• Do you know the Pythagorean triples 3-4-5 and 5-12-13?
• Can you find the area of a sector given the central angle and radius?
• Do you recall the side ratios for 30-60-90 and 45-45-90 triangles?
• Can you calculate distance and midpoint between two coordinate points?
• Do you know the equation of a circle and how to extract the center and radius from it?
• Can you apply SOH CAH TOA to find a missing side length?
• Do you remember that volume of a cone is \frac{1}{3} that of a cylinder?

Final Exam Pitfalls

1. The "Diameter vs. Radius" Trap: The question gives the diameter, but the formula requires radius (or vice versa). Always underline these words.
   • Correction: If d=10, write r=5 immediately.
2. Visual Assumptions: Assuming two lines are perpendicular because they look like it.
   • Correction: Never assume a right angle unless you see the square symbol or the text says "perpendicular" or "rectangle."
3. Coordinate Signs: Misidentifying the center of a circle from its equation.
   • Correction: For (x+2)^2 + (y-3)^2 , the center is (-2, 3), not (2, -3). The formula uses minus signs (x-h), so the signs flip.
4. Shadow/Similarity Errors: In similarity problems, matching the wrong sides.
   • Correction: Draw the triangles separately and orient them in the same direction before setting up your proportions.
5. Unit Confusion: Calculating an area in square feet when dimensions were given in inches.
   • Correction: Convert units before doing the calculation, or remember that 1 \text{ sq ft} = 144 \text{ sq inches}, not 12.

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GPT 5.2 Pro

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What You Need to Know

• Geometry on ACT Math is mostly about fast recognition—angle relationships, triangle facts, circle rules, and coordinate formulas.
• Memorize a small “toolkit” (distance/midpoint/slope, Pythagorean + special right triangles, circle formulas, area/volume formulas) and apply it under time pressure.
• Many questions are multi-step word problems—translate to a diagram/equation, then compute cleanly.

Curriculum basis: These notes align to ACT, Inc.’s published ACT Math content descriptions (Preparing for Higher Math—Geometry, plus related coordinate geometry and right-triangle trigonometry skills). The ACT does not publicly fix an exact percentage for each subtopic, so focus is based on the official topic list and common released-question patterns.

Congruence and Similarity

Congruence

Congruent figures have the same size and shape (all corresponding sides and angles match).

Common triangle congruence tests:

• SSS: three sides match
• SAS: two sides and included angle match
• ASA / AAS: two angles and a side match
• HL (right triangles): hypotenuse and one leg match

Key fact: If triangles are congruent, then CPCTC (corresponding parts of congruent triangles are congruent).

Similarity

Similar figures have the same shape but may differ in size.

• Corresponding angles equal
• Corresponding sides proportional

Similarity tests:

• AA: two angles match
• SAS (proportional): included angle equal and surrounding sides proportional
• SSS (proportional): all corresponding side ratios equal

Scale factor effects:

• Side lengths scale by k
• Areas scale by k^2
• Volumes scale by k^3

Worked example (similarity):
Two similar triangles have corresponding sides 6 and 9. If the smaller triangle’s area is 20, find the larger’s area.
1) Scale factor k=\frac{9}{6}=\frac{3}{2}
2) Area factor k^2=\left(\frac{3}{2}\right)^2=\frac{9}{4}
3) Larger area =20\cdot\frac{9}{4}=45

Exam Focus

• Why it matters: ACT uses similarity/congruence to test proportional reasoning and quick proof-like logic without formal proofs.
• Typical question patterns:
  • Identify a congruence/similarity criterion from a diagram and conclude a missing angle/side.
  • Use scale factor to compute a missing length (or area via k^2).
  • Set up proportions from corresponding sides.
• Common mistakes:
  • Mixing up corresponding sides—match vertices consistently.
  • Using k instead of k^2 for area (or k^3 for volume).
  • Assuming triangles are similar from one side ratio without angle info.

Properties of Triangles and Pythagorean Theorem

Triangle angle and side facts

• Interior angle sum: A+B+C=180^\circ
• Exterior angle equals sum of the two remote interior angles.
• Triangle inequality: for sides a,b,c, you need a+b>c, a+c>b, b+c>a.
• Isosceles triangle: equal sides imply equal base angles.

Pythagorean theorem and triples

For right triangle legs a,b and hypotenuse c:
a^2+b^2=c^2
Common triples:

• 3,4,5 (and multiples)
• 5,12,13 (and multiples)
• 8,15,17 (and multiples)

Special right triangles

• 45^\circ-45^\circ-90^\circ: legs x,x, hypotenuse x\sqrt{2}
• 30^\circ-60^\circ-90^\circ: short leg x, long leg x\sqrt{3}, hypotenuse 2x

Worked example (Pythagorean):
A ladder reaches a window 12 ft high with its base 5 ft from the wall. Ladder length?
c=\sqrt{12^2+5^2}=\sqrt{144+25}=\sqrt{169}=13 ft.

Exam Focus

• Why it matters: Triangles are the backbone of ACT geometry, especially right triangles in diagrams and word problems.
• Typical question patterns:
  • Find a missing side using a^2+b^2=c^2 or a known triple.
  • Use special-triangle ratios to avoid long algebra.
  • Use angle sum 180^\circ to find unknown angles.
• Common mistakes:
  • Using Pythagorean theorem on non-right triangles.
  • Forgetting which side is the hypotenuse (it’s opposite the 90^\circ angle).
  • Confusing 30–60–90 sides (short leg is opposite 30^\circ).

Properties of Circles

Core measures

For radius r and diameter d=2r:

• Circumference: C=2\pi r=\pi d
• Area: A=\pi r^2

Angles, arcs, chords, tangents

• Central angle measure equals its intercepted arc (in degrees).
• Inscribed angle equals half its intercepted arc.
• A tangent is perpendicular to the radius at the point of tangency.
• Equal chords subtend equal arcs (and vice versa).

A common ACT relationship with intersecting chords (inside a circle):
If two chords intersect, then a\cdot b=c\cdot d (products of segment lengths).

Worked example (inscribed angle):
An inscribed angle intercepts an arc of 110^\circ. Angle measure?
\text{angle}=\frac{110^\circ}{2}=55^\circ

Exam Focus

• Why it matters: Circle questions reward memorized relationships (half/double rules) and clean formula use.
• Typical question patterns:
  • Compute circumference/area from r or d.
  • Use inscribed vs central angle relationships.
  • Use tangent–radius perpendicular fact to find angles.
• Common mistakes:
  • Using diameter where radius is needed (or vice versa).
  • Forgetting the inscribed angle is half the intercepted arc.
  • Treating a tangent segment like a radius (it is not).

Surface Area and Volume of Solids

Most-tested formulas

Prism (any base):

• Volume: V=Bh where B is base area

Rectangular prism with dimensions l,w,h:

• V=lwh
• SA=2(lw+lh+wh)

Cylinder radius r, height h:

• V=\pi r^2h
• Total surface area: SA=2\pi r^2+2\pi rh

Pyramid:

• V=\frac{1}{3}Bh

Cone:

• V=\frac{1}{3}\pi r^2h

Sphere:

• V=\frac{4}{3}\pi r^3
• SA=4\pi r^2

Real-world tie-in: paint/wrapping uses surface area; filling tanks uses volume—units matter.

Worked example (cylinder volume):
A cylinder has r=3 and h=10. Volume:
V=\pi(3^2)(10)=90\pi

Exam Focus

• Why it matters: These are high-yield “plug-in” questions—fast points if formulas and units are solid.
• Typical question patterns:
  • Convert a word problem into V=Bh or a known solid formula.
  • Compare volumes/surface areas when dimensions scale by a factor k.
  • Mixed units (inches/feet) requiring conversion.
• Common mistakes:
  • Confusing surface area with volume.
  • Forgetting factors like \frac{1}{3} for cones/pyramids.
  • Unit errors (square units for area, cubic units for volume).

Trigonometric Ratios and Right Triangles

SOHCAHTOA

In a right triangle relative to an acute angle \theta:

• \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}
• \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}
• \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Useful identities (right-triangle context):

• \tan\theta=\frac{\sin\theta}{\cos\theta}
• Complement rule: \sin\theta=\cos(90^\circ-\theta)

Applications: angle of elevation/depression, shadow problems, ramps.

Worked example (angle of elevation):
A tree casts a 20 ft shadow. The angle of elevation to the top is 30^\circ. Height?
\tan 30^\circ=\frac{h}{20} \Rightarrow h=20\tan 30^\circ=20\cdot\frac{\sqrt{3}}{3}=\frac{20\sqrt{3}}{3}

Exam Focus

• Why it matters: ACT often mixes trig with geometry diagrams—one trig ratio plus Pythagorean theorem solves many problems.
• Typical question patterns:
  • Find a missing side using a given \sin/\cos/\tan value.
  • Use a special angle (like 30^\circ or 45^\circ) to simplify.
  • Translate a word scenario into “opposite/adjacent/hypotenuse.”
• Common mistakes:
  • Choosing the wrong “opposite” side (it depends on \theta).
  • Mixing up \sin and \cos when angle changes.
  • Rounding too early—keep exact values like \frac{\sqrt{3}}{2} when possible.

Equations of Conic Sections

ACT most commonly emphasizes recognizing/using standard forms (especially circles; parabolas show up in coordinate geometry contexts).

Circle

Center (h,k) and radius r:
(x-h)^2+(y-k)^2=r^2
If expanded, you may see: x^2+y^2+Dx+Ey+F=0.

Worked example (circle):
Circle with center (2,-1) and radius 5:
(x-2)^2+(y+1)^2=25

Parabola (common forms)

Vertical axis (vertex form):
y=a(x-h)^2+k
Horizontal axis:
x=a(y-k)^2+h
Key features: vertex (h,k) ; direction from sign of a.

Ellipse and hyperbola (less common, but possible)

Centered at (h,k) :

• Ellipse: \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1
• Hyperbola: \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 (or swapped)

Exam Focus

• Why it matters: Coordinate-form recognition is a speed skill—match an equation to its graph features (center/vertex/radius).
• Typical question patterns:
  • Identify center and radius from (x-h)^2+(y-k)^2=r^2 .
  • Convert between expanded and standard form (complete the square).
  • For y=a(x-h)^2+k, read vertex and opening direction.
• Common mistakes:
  • Sign errors: (x-h) means center at h (not -h).
  • Forgetting to take the square root when finding r from r^2.
  • Treating any quadratic as a parabola without checking variable structure.

Angles and Parallel Lines

Angle relationships

• Vertical angles are equal.
• Linear pair sums to 180^\circ.
• Complementary sum to 90^\circ.

With parallel lines cut by a transversal:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Same-side (consecutive) interior angles sum to 180^\circ.

Polygons (sometimes grouped with angle skills)

For an n-gon:

• Sum of interior angles: (n-2)\cdot 180^\circ
• Each interior angle (regular): \frac{(n-2)\cdot 180^\circ}{n}

Worked example (parallel lines):
If corresponding angles are labeled (3x+10)^\circ and (5x-30)^\circ , set equal:
3x+10=5x-30 \Rightarrow 40=2x \Rightarrow x=20

Exam Focus

• Why it matters: These are “diagram algebra” questions—quick angle equations yield fast points.
• Typical question patterns:
  • Solve for x using a pair of equal angles (corresponding/alternate interior/vertical).
  • Use supplementary relationships to form an equation summing to 180^\circ.
  • Compute polygon interior angle sums.
• Common mistakes:
  • Mixing up which angles are equal vs supplementary.
  • Forgetting the lines must be parallel for corresponding/alternate interior facts.
  • Using regular-polygon formula on a non-regular polygon.

Coordinate Geometry

Core formulas

Slope between points (x1,y1) and (x2,y2) :
m=\frac{y2-y1}{x2-x1}
Distance:
d=\sqrt{(x2-x1)^2+(y2-y1)^2}
Midpoint:
\left(\frac{x1+x2}{2},\frac{y1+y2}{2}\right)
Line forms:

• Slope-intercept: y=mx+b
• Point-slope: y-y1=m(x-x1)
  Parallel/perpendicular:
• Parallel lines have equal slopes.
• Perpendicular (non-vertical) slopes satisfy m1m2=-1.

Coordinate geometry of shapes

• Right angle check: slopes perpendicular (product -1).
• Parallel sides: equal slopes.
• Use distance to check equal side lengths (e.g., isosceles triangles, squares).

Worked example (distance):
Find distance between (-1,2) and (3,5):
d=\sqrt{(3-(-1))^2+(5-2)^2}=\sqrt{4^2+3^2}=5

Exam Focus

• Why it matters: ACT loves coordinate “toolkit” problems—plug in, simplify, and interpret.
• Typical question patterns:
  • Find slope/distance/midpoint quickly.
  • Write an equation of a line through a point with given slope.
  • Identify geometric properties (parallel, perpendicular, equal lengths) using slopes/distances.
• Common mistakes:
  • Sign errors in slope (order of subtraction must match in numerator/denominator).
  • Forgetting vertical lines have undefined slope and equation x=c.
  • Dropping the square root when computing distance.

Quick Review Checklist

• Can you decide triangle congruence using SSS, SAS, ASA, AAS, or HL?
• Can you set up a similarity proportion and use k^2 for area scaling?
• Do you know a^2+b^2=c^2 and the 45–45–90 and 30–60–90 ratios?
• Can you use C=2\pi r and A=\pi r^2 and the inscribed-angle “half-arc” rule?
• Can you compute V and SA for prisms, cylinders, cones/pyramids, and spheres (including factors like \frac{1}{3} and \frac{4}{3})?
• Can you apply \sin\theta, \cos\theta, \tan\theta correctly from a right-triangle diagram?
• Can you read center/radius from (x-h)^2+(y-k)^2=r^2 and vertex from y=a(x-h)^2+k ?
• Can you solve angle-chasing problems with parallel lines (corresponding/alternate interior/supplementary)?
• Can you compute slope, distance, and midpoint and write y=mx+b or y-y1=m(x-x1)?

Final Exam Pitfalls

1. Using the wrong “version” of a formula (radius vs diameter; area vs circumference; surface area vs volume) — pause and label what the problem is actually asking.
2. Matching the wrong corresponding parts in similarity/congruence — keep vertex order consistent and mark angles/sides before writing proportions.
3. Pythagorean theorem on a non-right triangle — confirm a 90^\circ angle (or perpendicular lines) before using a^2+b^2=c^2.
4. Trig ratio misidentification — define “opposite/adjacent” relative to the given \theta, not relative to the page.
5. Coordinate sign/slope mistakes — use a consistent subtraction order for m and carefully square differences in the distance formula.

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Claude Opus 4.6

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What You Need to Know

• Geometry accounts for roughly 23–28% of the ACT Math section (about 14–16 questions out of 60). Mastering core properties of triangles, circles, angles, and coordinate geometry is essential for a strong score.
• The ACT provides no formula sheet. You must memorize key formulas — Pythagorean theorem, area and volume formulas, trig ratios, circle equations, and slope/distance/midpoint formulas.
• Most geometry questions are application-based: you'll be given a figure or scenario and asked to find a missing length, angle, area, or coordinate. Multi-step reasoning is common.
• Coordinate geometry and trigonometry questions tend to appear in the later (harder) portion of the test, while basic angle and triangle questions appear earlier.

Congruence and Similarity

Congruent figures have the same shape and size — all corresponding sides and angles are equal. Similar figures have the same shape but may differ in size — corresponding angles are equal, and corresponding sides are in proportion.

Key Congruence Tests (Triangles)

• SSS — three pairs of equal sides
• SAS — two sides and the included angle
• ASA — two angles and the included side
• AAS — two angles and a non-included side
• HL — hypotenuse-leg (right triangles only)

Similarity Criteria

• AA — two pairs of equal angles
• SSS similarity — all three pairs of sides in proportion
• SAS similarity — two sides in proportion with the included angle equal

When figures are similar with scale factor k, corresponding sides satisfy \frac{a1}{a2} = k, areas scale by k^2, and volumes scale by k^3.

Exam Focus

• Why it matters: 2–3 questions typically involve proportion setups using similar triangles or identifying congruence.
• Typical question patterns:
  • A diagram with two overlapping or nested triangles; find a missing side using proportions.
  • Word problems involving shadows, scale models, or maps that require setting up a similarity ratio.
  • Identifying which congruence/similarity criterion applies.
• Common mistakes:
  • Setting up proportions incorrectly — always match corresponding sides (shortest to shortest, etc.).
  • Confusing congruence with similarity; congruence requires equal size, not just equal shape.
  • Forgetting the area/volume scaling rules (using k instead of k^2 for area).

Properties of Triangles and the Pythagorean Theorem

The angle sum of any triangle is 180°. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c:
a^2 + b^2 = c^2

Special Right Triangles

Common Pythagorean triples worth memorizing: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples).

Triangle Inequality

The sum of any two sides must be greater than the third side.

Area Formula

A = \frac{1}{2}bh

Exam Focus

• Why it matters: Right triangles and the Pythagorean theorem appear in 3–5 questions and underpin trig and distance problems.
• Typical question patterns:
  • Finding a missing side of a right triangle embedded in a real-world scenario (ladders, ramps, diagonals).
  • Using special right triangle ratios to bypass lengthy calculations.
  • Multi-step problems combining triangle properties with area or perimeter.
• Common mistakes:
  • Applying the Pythagorean theorem to non-right triangles.
  • Misidentifying the hypotenuse — it's always opposite the 90° angle and is the longest side.
  • Mixing up the 30-60-90 ratios (the side opposite 30° is the shortest, not \sqrt{3}).

Properties of Circles

Key definitions: radius (center to edge), diameter (d = 2r), chord (segment with both endpoints on the circle), tangent (line touching the circle at exactly one point — perpendicular to the radius at that point), secant (line intersecting the circle at two points).

Essential Formulas

• Circumference: C = 2\pi r
• Area: A = \pi r^2
• Arc length: s = \frac{\theta}{360} \times 2\pi r
• Sector area: A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2

Angle Relationships

• A central angle equals its intercepted arc.
• An inscribed angle equals half its intercepted arc.
• An angle inscribed in a semicircle is 90°.

Exam Focus

• Why it matters: Circle problems appear 2–4 times and often combine arc/sector calculations with algebra.
• Typical question patterns:
  • Finding arc length or sector area given a central angle.
  • Using the inscribed angle theorem to find missing angles.
  • Tangent-radius perpendicularity used to create a right triangle.
• Common mistakes:
  • Confusing diameter with radius in formulas (using d in \pi r^2 without halving).
  • Forgetting to convert between degrees and the fraction of a full circle.
  • Assuming an inscribed angle equals the arc (it's half).

Surface Area and Volume of Solids

Exam Focus

• Why it matters: 1–3 questions involve 3D solids, often in word-problem form.
• Typical question patterns:
  • "A cylinder has radius 5 and height 10. What is its volume?"
  • Composite solids — e.g., a hemisphere on top of a cylinder.
  • Comparing volumes when dimensions are scaled.
• Common mistakes:
  • Forgetting the \frac{1}{3} factor for cones and pyramids.
  • Confusing slant height with vertical height for cones.
  • Using diameter instead of radius.

Trigonometric Ratios and Right Triangles

Use SOH-CAH-TOA as your mnemonic:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Key Values to Know

The ACT occasionally tests the Law of Sines and Law of Cosines on harder questions:
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
c^2 = a^2 + b^2 - 2ab\cos C

Exam Focus

• Why it matters: 2–4 questions directly test trig — these tend to be mid-to-high difficulty.
• Typical question patterns:
  • Given a right triangle with one side and one acute angle, find another side.
  • Angle of elevation/depression word problems.
  • Using inverse trig to find an angle.
• Common mistakes:
  • Mixing up opposite and adjacent sides relative to the given angle.
  • Using trig ratios on non-right triangles without the Law of Sines/Cosines.
  • Calculator in radian mode instead of degree mode.

Equations of Conic Sections

The ACT primarily tests circles and occasionally ellipses or parabolas in the coordinate plane.

Circle (Standard Form)

(x - h)^2 + (y - k)^2 = r^2
Center: (h, k), Radius: r

Ellipse

\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

Parabola

Vertex form: y = a(x - h)^2 + k

You may need to complete the square to convert a general equation into standard form.

Example: Convert x^2 + y^2 - 6x + 4y - 12 = 0 to standard form.

1. Group: (x^2 - 6x) + (y^2 + 4y) = 12
2. Complete the square: (x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4
3. Result: (x - 3)^2 + (y + 2)^2 = 25 → Center (3, -2), radius 5

Exam Focus

• Why it matters: 1–2 questions test conic sections, usually circles; these are often in the last 20 questions.
• Typical question patterns:
  • Identify center and radius from a circle equation (sometimes requiring completing the square).
  • Determine if a point lies on, inside, or outside a circle.
  • Match an equation to a described geometric figure.
• Common mistakes:
  • Sign errors when reading center from (x - h)^2 + (y - k)^2 = r^2 — if the equation has (x + 3), then h = -3.
  • Forgetting to add the completed-square constants to both sides.
  • Confusing r^2 with r (if r^2 = 25, the radius is 5, not 25).

Angles and Parallel Lines

When a transversal crosses two parallel lines, it creates eight angles with special relationships:

Vertical angles (formed by any two intersecting lines) are always equal. Supplementary angles sum to 180°; complementary angles sum to 90°.

The sum of interior angles of a polygon with n sides:
(n - 2) \times 180°

Exam Focus

• Why it matters: 2–3 questions test angle relationships, often among the easier questions on the test.
• Typical question patterns:
  • A diagram with parallel lines cut by a transversal; find a labeled angle.
  • Using angle relationships plus algebra (e.g., 3x + 10 = 5x - 20).
  • Interior angle sums in polygons.
• Common mistakes:
  • Confusing alternate interior with co-interior angles — one pair is equal, the other is supplementary.
  • Assuming lines are parallel without it being stated.
  • Arithmetic errors in the polygon angle formula.

Coordinate Geometry

Core Formulas

• Distance: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
• Midpoint: M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)
• Slope: m = \frac{y2 - y1}{x2 - x1}
• Slope-intercept form: y = mx + b
• Point-slope form: y - y1 = m(x - x1)

Key Slope Relationships

• Parallel lines have equal slopes: m1 = m2
• Perpendicular lines have negative reciprocal slopes: m1 \cdot m2 = -1

Example: Find the distance between (2, -1) and (5, 3).
d = \sqrt{(5-2)^2 + (3-(-1))^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Exam Focus

• Why it matters: Coordinate geometry appears in 3–5 questions spanning easy to hard difficulty.
• Typical question patterns:
  • Find the midpoint or distance between two points.
  • Determine the equation of a line parallel or perpendicular to a given line through a specific point.
  • Identify which quadrant a point or line segment lies in.
• Common mistakes:
  • Sign errors in the distance formula — always subtract in the same order or square the result (negatives cancel).
  • Confusing negative reciprocal with just the negative (perpendicular to m = 2 is m = -\frac{1}{2}, not m = -2).
  • Forgetting that a vertical line has an undefined slope and a horizontal line has slope 0.

Quick Review Checklist

• ☐ Can you state and apply the Pythagorean theorem and recognize common triples?
• ☐ Do you know the side ratios for 30-60-90 and 45-45-90 triangles?
• ☐ Can you set up and solve proportions using similar triangles?
• ☐ Do you know the formulas for circumference, arc length, area, and sector area of a circle?
• ☐ Can you calculate the volume and surface area of cylinders, cones, spheres, and prisms?
• ☐ Can you use SOH-CAH-TOA to find missing sides or angles in a right triangle?
• ☐ Can you convert a circle equation to standard form by completing the square?
• ☐ Do you know the angle relationships formed by parallel lines and a transversal?
• ☐ Can you find slope, distance, and midpoint from two coordinate points?
• ☐ Can you write the equation of a line parallel or perpendicular to a given line?

Final Exam Pitfalls

1. No formula sheet: The ACT does not provide formulas. Arrive with area, volume, trig ratio, and coordinate geometry formulas fully memorized — don't waste time trying to derive them.
2. Radius vs. diameter mix-up: Many problems give the diameter. Always check whether you need to halve it before plugging into \pi r^2 or 2\pi r.
3. Forgetting the \frac{1}{3} in cone/pyramid volume: Students routinely calculate \pi r^2 h instead of \frac{1}{3}\pi r^2 h. If the solid tapers to a point, include the fraction.
4. Sign errors in circle equations: (x + 3)^2 means the center's x-coordinate is -3, not +3. Read the signs carefully.
5. Perpendicular slope errors: The perpendicular slope is the negative reciprocal — flip the fraction and change the sign. Students often only do one of those two operations.
6. Using trig on non-right triangles: SOH-CAH-TOA only works for right triangles. For oblique triangles, you need the Law of Sines or Law of Cosines.