Geometry A Cheat Sheet Formulas

Midpoint Formula

• Formula: ((x_1+x_2/2, y_1+y_2/2))
• Used to find the point that divides a line segment into two equal parts.

Distance Formula

• Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
• Calculates the straight line distance between two points in a Cartesian plane.

Slope Formula

• Formula: (m = \frac{y_2 - y_1}{x_2 - x_1})
• Determines the steepness or incline of a line, comparing changes in y and x.

Line Equation

• Formula: (y = mx + b)
• Represents a straight line where (m) is the slope and (b) is the y-intercept.

Reflection over x-axis

• Transformation: ((x, y) \rightarrow (x, -y))
• Inverts the y-coordinate, reflecting points across the x-axis.

Reflection over y-axis

• Transformation: ((x, y) \rightarrow (-x, y))
• Inverts the x-coordinate, reflecting points across the y-axis.

Reflection over y=x

• Transformation: ((x, y) \rightarrow (y, x))
• Swaps x and y coordinates, reflecting points across the line (y = x).

Reflection over y=-x

• Transformation: ((x, y) \rightarrow (-y, -x))
• Inverts both coordinates, reflecting points across the line (y = -x).

Reflection over origin

• Transformation: ((x, y) \rightarrow (-x, -y))
• Inverts both coordinates, reflecting points through the origin.

Reflection over h & k

• Transformation: ((x, y) \rightarrow (x + h, y + k))
• Moves points horizontally by (h) and vertically by (k).

Translation of h & k

• Definitions: (h) indicates horizontal movement, (k) indicates vertical movement.
• Shifts points in the Cartesian plane without changing their orientation.

Rotate 90 degrees

• Transformation: ((x, y) \rightarrow (-y, x))
• Rotates points 90 degrees counter-clockwise around the origin.

Rotate 180 degrees

• Transformation: ((x, y) \rightarrow (-x, -y))
• Rotates points 180 degrees, resulting in a direct inversion.

Rotate 270 degrees

• Transformation: ((x, y) \rightarrow (y, -x))
• Rotates points 270 degrees counter-clockwise or 90 degrees clockwise.

Rotate -90 degrees

• Transformation: ((x, y) \rightarrow (y, -x))
• Equivalent to rotating 270 degrees counter-clockwise.

Rotate -180 degrees

• Transformation: ((x, y) \rightarrow (-x, -y))
• Same effect as rotating 180 degrees.

Rotate -270 degrees

• Transformation: ((x, y) \rightarrow (-y, x))
• Rotates points 90 degrees clockwise.

Dilation

• Transformation: ((x, y) \rightarrow (kx, ky))
• Scales points from the origin by a factor (k), altering the size while maintaining shape.

Sine (SOH)

• Relationship: (\sin A= \frac{a}{c}, \sin B= \frac{b}{c})
• Inverse sine function: (\sin^{-1}) used to find angle measures.

Cosine (CAH)

• Relationship: (\cos A= \frac{b}{c}, \cos B= \frac{a}{c})
• Inverse cosine function: (\cos^{-1}) utilized for angle calculations.

Tangent (TOA)

• Relationship: (\tan A= \frac{a}{b}, \tan B= \frac{b}{a})
• Inverse tangent function: (\tan^{-1}) helps to determine angle measures.

Pythagorean Theorem

• Formula: (a^2 + b^2 = c^2)
• Relates the lengths of the sides of a right triangle, where (c) is the hypotenuse.

Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Used to find the roots of a quadratic equation in the form (ax^2 + bx + c = 0).

This quiz covers essential formulas used in Geometry A, including the midpoint formula, distance formula, slope formula, and line equations. Test your understanding of how to apply these formulas in various geometric contexts with flashcard-style questions.