Lesson 8 Notes
Table of Contents
- Lines
- Transformations
- Equations of Circles
- Inscribed Angles & Central Angles (in Circles)
- Cyclic Quadrilaterals
Lines
- A line is parallel to another line if they have the same slope and a different y-intercept
- A line is perpendicular to another line if the slopes of the two lines are negative reciprocols of one-another.
Examples
In class examples
Transformations
Transformations can be noted with the following notation:
- (x, y) -> (x + 2, y - 1) = a translation where you slide the pre-image to the image 2 units right and 1 unit down.
- (x, y) -> (−x, −y) = a rotation where you rotate a pre-image to the image 180°
Equations of Circles
Circles have the equation (x - a)2 + (y - b)2 = r2
Examples
- Graph (x - 2)2 + (y - 4)2 = 36
- Graph x2 + y2 = 1
- Graph (x + 3)2 + (y + 1)2 = 4
Inscribed & Central Angles in Circles
Inscribed Angle - an angle made from points sitting on the circle's circumference.
Central Angle - an angle where the vertex of the angle is at the center of the circle and the arms extend to the circumference of a circle.
Circle Theoremes
- Angles Subtended by the Same Arc Theorem - an inscribed angle whose arms cover the same arc (see notes in class) has the same angle measure as another inscribed angle whose arms cover the same arc.
- Angle at the Center Theorem - a central angle whose arms cover a certain arc (in class drawings) has half the measure of the inscribed angle covering the same arc.
- Thales' Theorem (An Angle in a Semicircle) - An inscribed angle across a circles diameter is always 90°.
Cyclic Quadrilateral
A Cyclic Quadrilateral has every vertex on the circle's circumference.
A Cyclic Quadrilateral's opposite angles add to 180° (see in class notes).
Tangent Angle
A tangent line touches the circle at just one point. The angle formed by the tangent line and the circle's radius is always 90°