📚SAT (Digital) Unit 4 – SAT Math – Geometry and Trigonometry

Key Concepts

• Understand the properties of various geometric shapes (triangles, circles, rectangles)
• Know how to calculate the area and perimeter of different shapes
  • Area of a triangle: $A = \frac{1}{2}bh$
  • Area of a circle: $A = \pi r^2$
• Familiarize yourself with the Pythagorean theorem: $a^2 + b^2 = c^2$
• Grasp the concepts of parallel and perpendicular lines
• Learn the basics of trigonometric functions (sine, cosine, tangent)
• Understand how to use the coordinate plane to solve geometric problems
• Know how to apply geometric concepts to real-world situations

Geometric Shapes and Properties

• Triangles have three sides and three angles that sum to 180°
  • Equilateral triangles have three equal sides and three equal angles (60°)
  • Isosceles triangles have two equal sides and two equal angles
• Circles are defined by their radius and have a constant diameter (twice the radius)
• Rectangles have four right angles and opposite sides of equal length
  • Squares are a special case of rectangles with all sides equal
• Parallelograms have opposite sides parallel and equal in length
• Trapezoids have one pair of parallel sides
• Similar shapes have the same shape but different sizes, with proportional side lengths
• Congruent shapes have the same shape and size, with equal side lengths and angles

Coordinate Geometry

• The coordinate plane consists of two perpendicular axes (x-axis and y-axis)
• Points on the coordinate plane are represented by ordered pairs (x, y)
• The distance between two points can be calculated using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• The midpoint of a line segment can be found using the midpoint formula: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
• Slope represents the steepness of a line and can be calculated using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
  • Parallel lines have the same slope
  • Perpendicular lines have slopes that are negative reciprocals of each other
• The equation of a line can be written in slope-intercept form: $y = mx + b$

Trigonometry Basics

• Trigonometry deals with the relationships between the sides and angles of triangles
• The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan)
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• The reciprocal functions are cosecant (csc), secant (sec), and cotangent (cot)
• Trigonometric identities, such as the Pythagorean identity ($\sin^2 \theta + \cos^2 \theta = 1$), can help simplify expressions
• The unit circle is a circle with a radius of 1 centered at the origin, used to define trigonometric functions
• Radians are an alternative unit for measuring angles, where $2\pi$ radians equal 360°

Problem-Solving Strategies

• Read the question carefully and identify the given information and what you need to find
• Draw a diagram or sketch to visualize the problem, labeling known values
• Break down complex problems into smaller, manageable steps
• Use formulas and theorems appropriately, substituting given values
• Double-check your work and ensure your answer makes sense in the context of the problem
• Practice solving a variety of problems to develop your problem-solving skills
  • Focus on understanding the underlying concepts rather than just memorizing formulas
• Look for patterns or similarities between problems to help you solve new ones more efficiently

Common Mistakes to Avoid

• Misreading or misinterpreting the question
• Confusing similar-sounding terms or formulas (e.g., area vs. perimeter)
• Forgetting to label diagrams or mixing up labels
• Making calculation errors due to rushing or lack of attention to detail
  • Use a calculator when permitted, but be careful to input values correctly
• Applying formulas or theorems incorrectly or in the wrong context
• Neglecting to consider special cases or exceptions to rules
• Providing answers in the wrong units or forgetting to include units altogether
• Not checking your work for consistency and reasonableness

Practice Questions

• Find the area of a right triangle with a base of 6 cm and a height of 8 cm
• Calculate the circumference of a circle with a diameter of 10 inches
• Determine the length of the hypotenuse of a right triangle with legs measuring 3 units and 4 units
• Given points A(2, 3) and B(5, 7), find the midpoint of line segment AB
• Calculate the slope of the line passing through points (1, 2) and (4, 8)
• If $\sin \theta = \frac{3}{5}$, find the value of $\cos \theta$
• A ladder 10 feet long leans against a wall, forming a 60° angle with the ground. How high up the wall does the ladder reach?

Tips and Tricks

• Memorize common formulas and theorems to save time during the exam
  • Create flashcards or use mnemonic devices to help you remember
• Use the process of elimination to narrow down answer choices in multiple-choice questions
• When faced with a complex problem, start by solving a simpler, related problem and then build upon that solution
• Look for right angles in diagrams, as they can often help you apply the Pythagorean theorem
• Remember that the sum of the angles in a triangle is always 180°, which can help you find missing angle measures
• If you're stuck on a problem, try approaching it from a different perspective or using an alternate method
• Manage your time effectively during the exam, moving on from difficult problems and returning to them later if time allows