SAT Math: Geometry and Trigonometry Explained

Welcome to SAT Geometry Essentials! Today we'll explore fundamental shapes and their properties.Let's start with rectangles, one of the most common shapes on the SAT.For rectangles, remember these key formulas: Area equals width times height, and perimeter equals two times width plus height.Triangles are crucial for the SAT. The area is one-half base times height, and perimeter is the sum of all sides.The thirty-sixty-ninety triangle is a special right triangle with unique side ratios.If the shortest side is x, then the hypotenuse is twice that, and the remaining side is x times the square root of three.The forty-five-forty-five-ninety triangle is another special right triangle. Its two legs are equal, and its hypotenuse is the leg length times the square root of two.Circles are defined by their radius. The area is pi r squared, and the circumference is two pi r.Regular polygons have equal sides and angles. The sum of interior angles depends on the number of sides.For a polygon with n sides, the sum of interior angles is one hundred eighty degrees times n minus two.On the coordinate plane, we can find the distance between two points using the distance formula.The distance formula uses the difference in x and y coordinates to find the length of the line segment.The midpoint formula finds the point exactly halfway between two points.The slope of a line measures its steepness and direction.Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.Now let's explore geometric transformations starting with reflections.A rotation turns a figure around a fixed point by a specific angle.Translation moves every point of a figure the same distance in the same direction.In right triangles, trigonometric ratios help us find missing sides and angles.The three main ratios are sine, cosine, and tangent. Each relates two sides of the right triangle.These ratios are closely connected to the unit circle, which has a radius of one unit.As we move around the circle, we can see how sine and cosine values change.The most common angles tested on the SAT are thirty, forty-five, sixty, and ninety degrees.Let's solve a practical problem: finding the height of a building using the angle of elevation.Using tangent, which is opposite over adjacent, we can set up the equation: tangent of thirty degrees equals height over fifty meters.Solving for height, we multiply fifty by the tangent of thirty degrees.The building is approximately twenty-eight point eight seven meters tall.Let's review the key points for SAT trigonometry success.Keep practicing these concepts, and you'll be well-prepared for the SAT!