Unit 4 - Linear Geometry Flashcards
The Cartesian Plane
- Quadrants +/-
- Origin
- X and Y
The Cartesian plane is divided into four quadrants:
1️⃣ First Quadrant (Q1): (-, +) → x is negative, y is positive.
2️⃣ Second Quadrant (Q2): (+, +) → Both x and y are positive.
3️⃣ Third Quadrant (Q3): (-, -) → Both x and y are negative.
4️⃣ Fourth Quadrant (Q4): (+, -) → x is positive, y is negative.
- The ‘origin’ is at the centre (0,0)
- The position of any point is
specified as (x , y): x is the horizontal from the origin (x-coordinate), and y is the vertical from the origin (y-coordinate)
Pythagorus theorem
Can use Pythagoras to solve the distance, after forming a triangle using the rise/run, or when given a triangle: a^2 + b^2 = c^2
Midpoints
Xmidpoint = (x1+x2)/2
Ymidpoint = (y1+y2)/2
- Substitute in coordinates to find midpoint, or midpoint and a coordinate to find other coordinate
Gradients
- Slope/steepness of line
- Slope = (y - step)/(x-step) = (y2-y1)/(x2-x1)
- Note: When finding the gradient from a graph, if a point is in between, use a definitive one, as the slope is the same at any point in the line
Triangles (3 points)
- Calculate distance of each side (will show type of triangle, equalateral - all sides same, isoceles - two sides same, scalene - no sides same)
- Do pythagoreus to see if right angled
- Find the gradient of all three lines, note: if two lines are opposite to one another, they hit each other at a right angle)
- Find the midpoint if requested
Parallel lines
Rules:
1. If two lines are parallel, they must have the same gradient
2. If two lines have the same gradient, they must be parallel
Symbol: 1/ (1 over line)
Perpendicular lines
- Form a 90 degree angle between them
Rules:
1. If the lines are perpendicular, then their gradients are negative reciprocals, (opposite fraction and negative/positive)
2. If the gradients are negative reciprocals, then the lines are perpendicular
Symbol: //
Combining parallel and perpendicular
- Calculate gradient of line with both coordinates
- Flip to make negative reciprocal (which is the gradient of the other line, if they are perpendicular)
- Solve for unknown coordinate
Collinear
- 3 or more points are collinear if they lie on the same straight line
(A,B, and C are collinear if gradient of AB and gradient of BC are equal)
Slope Intercept Form
General Form
Slope int: y = mx+b
- Where m = slope/gradient
- b = y intercept
General: Ax+By=C
- Where A always has a positive coefficient
- There should be no fractions
Finding equation of line:
Info needed - gradient and at least one point, two points which lie on line
Use info to form slope int form, then convert to general if needed
Perpendicular Bisectors
- Find original slope. Convert to its perpendicular gradient.
- Use the slope intercept form. Input new slope.
- Use known point. Find B.
- Compile together. (Therefore line 1/ [perpendicular symbol] to [original line in slope int] is [second line in slope int])
Circles (equation of tangent of circle)
- Slope of line 1 (y2-y1)/(x2-x1), then do negative reciprocal to get slope of line 2
- Solve for equation of line and then b
