Week 4 Math - Geometry 2
Slope
"Slant" of a line. Rise over run. Difference between y coordinates/distance between x coordinates = (y2-y1)/(x2-x1)
Tangent
A line in the plane of a circle that intersects the circle in exactly one point. The point is the "point of tangency." A radius drawn to a point of tangency intersects that tangent at a 90 degree angle. If a problem contains a tangent draw a radius to the point of tangency and denote the 90 degree angle!
Radius
A line segment from the center to the circumference of a circle or sphere CANNOT assume a line segment intersects the center unless the question says so!
Graphing horizontal and vertical lines
NOT expressed in form y = mx + b! Horizontal lines run indefinitely but never rise, so slop = 0 (rise/run = 0/infinity). Expressed as y = b. Vertical lines have an UNDEFINED slope, as they rise indefinitely but never run (rise/run = infinity/0). Expressed as x = a, where a is the x-intercept.
Cones, pyramids & spheres
No formulas need to be known! DO know what terms refer to! cone = three-dimensional object that has a circular base and sides that taper to a point right circular cone = cone with apex that runs perpendicular to its base cross-section of a cone (horizontal slice) is always a CIRCLE! pyramid = three-dimensional object with a polygon for a base and triangular sides with a common vertex sphere = Three-dimensional circle. Distance from a point on the surface to the center is a radius. Every cross-section is a circle. Cross-sections toward the middle form larger circles than toward the surface. hemisphere = half a sphere
Three dimensional shape
Object that has height, width, and depth
Sector
Portion of the area of a circle that extends from the CENTER! Area determined by obtaining the area of a circle and its central angle (whose vertex lies at the center of the circle). Sector = (πr²)(x/360) (Customary to leave π as part of the answer.)
Arc
Portion of the circumference of a circle A MAJOR arc is greater than half the circumference. A MINOR arc is less than half the circumference.
Volume
Quantity of material that a three-dimensional object can hold. Calculated in terms of CUBIC units.
Intersection of two lines
Set the two equations equal to one another and solve for x. This will be the x coordinate of the point of intersection. Plug x into one of the initial equations and solve for y. This will be the y coordinate of the point of intersection.
Graphing equations
y=mx + b, where m denotes slope and b denotes the y intercept. CRITICAL in coordinate geometry to rewrite equations in the form y = mx +b! Enables graphing! Positive m indicates ascending line; negative m indicates descending. Positive b indicates y-intercept above the origin, negative below. Equations without a b term have the ORIGIN as its y-intercept. Equations without m term have slope 1.
When working with CIRCLES...
...label your RADII!
To establish if a line and point intersect
...plug coordinate values of the point into the equation of the line! If coordinates produce a true statement, they DO intersect! If not, they don't!
Coordinate plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis. Lines meet at the origin. Divided into four quadrants. Points have x and y coordinates. A point is identified by an ordered pair (x,y).
Diameter
A line segment passing from point to point through the center of a circle or sphere. The longest line segment that can be drawn between points on the circle. Twice the length of the radius. CANNOT assume a line segment intersects the center unless the question says so!
Chord
A segment whose endpoints lie on a circle. Can never be longer than the diameter! A chord running perpendicular to a radius will be bisected by that radius.
Cube
A three-dimensional shape having six congruent square faces. Each composed of four sides, traditionally labeled "s". Only ONE side is needed to determine surface area. Volume = s^3 Surface area = 6s^2 Diagonal of a cube = s √3
Area of a circle
A=πr², where r=radius and π~3.14 Only info needed is the radius! Customary to leave π as part of the answer! If question asks for "approximate" answer, use 3.1 or 22/7 for π!
Circle
An infinite number of points connected around an equal distance from a middle point`
Inscribed angle
Angle whose vertex lies on the perimeter of a circle. THREE special properties: 1) Half the measure of the central angle. 2) Inscribed angles drawn to the same arc have equal measures. 3) Inscribed angles of equal measure have arcs of equal length; arcs of equal length have inscribed angles of equal measure.
Circumference of a circle
C = πd = 2πr Only info needed is the radius! Customary to leave π as part of the answer! If question asks for "approximate" answer, use 3.1 or 22/7 for π!
Surface area
Collective space on the surface of a three-dimensional object. Calculated in terms of SQUARE units.
Rectangular solid
Composed of six rectangular faces. A side of a face is an edge. Each point at which two edges meet is a vertex. Surface area = sum of areas of faces
From two points to y = mx + b
Given two points, the equation of a line that passes through them can be determined. First establish slope using the slope formula. Then substitute the slope into the equation. Plug coordinates of on point in and solve for b.
Coordinate geometry problems
If unsure how to proceed: Establish the equation of any line possible. Plug points given into that equation or any equation(s) provided. CAN assume line runs through the origin if depicted as such, so (0,0) is a point and o = x and y intercepts!
Length of an arc
Length determined by obtaining the circumference and it's central angle (angle whose vertex lies at the center of the circle). Arc = (2πr)(x/360)
Coordinate plane: Distance and Midpoints
Length of a line between two points is the distance between them. The point halfway between is the midpoint. To find the distance, consider the two points the hypotenuse of a right triangle. Calculate the legs by measuring the rise and run. Look for the special right triangle or use Pythagorean Theorem to determine the hypotenuse length. Midpoint coordinates are the average of x and y coordinates of the two points = ((x1 +x2)/2 , (y1 +y2)/2)
Finding x-intercept
Let y=0; solve y = mx + b
Graphing parallel and perpendicular lines
Lines in the same plane that do not intersect are parallel. Lines that intersect at a 90 degree angle are perpendicular. Parallel lines have the same slope! Perpendicular lines have slopes whose products = -1!
Cylinders
Two circles and a rectangle joined together. Volume = πr^2h Surface area = 2πrh + 2πr^2 (rectangle + two circles) Surface area of a band of a cylinder is just part of the rectangle = 2πr(height of the band)
With arcs and sectors
You can always determine one of three unknowns--central angle, radius, or sector.
Diagonal of a rectangular solid
d = √(l^2 + w^2 + h^2) Partial diagonals are determined in the same manner as full diagonals!
