Coordinate Geometry

Question 1

Slope Formula

Answer 1

m = (y₂ - y₁) / (x₂ - x₁)

Question 2

Lines with Positive Slope

Answer 2

• A positive slope line will always intersect quadrants I and III, but not necessarily II or IV.
• x-intercept > 0 → intersects quadrant I, III, and IV
• x-intercept < 0 → intersects quadrant I, III, and II
• x-intercept = 0 → only intersect quadrant I and III

Question 3

Lines with Negative Slope

Answer 3

• A negative slope line will always intersect quadrants II and IV, but not necessarily I or III.
• x-intercept > 0 → intersects quadrant II, IV, and I
• x-intercept < 0 → intersects quadrant II, IV, and III
• x-intercept = 0 → only intersect quadrant II and IV
• The x- and y-intercepts of a negatively sloped line have the same sign

Question 4

Lines with 0 Slope (Horizontal Lines)

Answer 4

• Slope = 0
• y-intercept > 0 → intersects quadrants I, II
• y-intercept < 0 → intersects quadrants III, IV
• y-intercept = 0 → intersects no quadrant

Question 5

Lines with Undefined Slope (Vertical Lines)

Answer 5

• Slope = Undefined
• x-intercept > 0 → intersects quadrants I, IV
• x-intercept < 0 → intersects quadrant II, III
• x-intercept = 0 → intersects no quadrants

Question 6

Slope and Steepness of a Line

Answer 6

The larger the absolute value of the slope of a line, the steeper the line.

Question 7

Slope-Intercept Equation

Answer 7

y = mx + b

Question 8

Point Slope Form

Answer 8

y - y₁ = m(x - x₁)

• y₁ = y coordinate of one point
• x₁ = x coordinate of one point
• Useful when one point is given
• Essentially the rearranged slope formula!

Question 9

Equations for Horizontal and Vertical Lines

Answer 9

• Vertical line equation: x = a
• Horizontal line equation: y = b

Question 10

Standard Form of the Equation of a Line

Answer 10

Ax + By = C

A, B, C are all constants

Covert it to the slope-intercept form

Question 11

Perpendicular Lines

Answer 11

The slope of perpendicular lines form negative reciprocals, i.e. m₁m₂ = -1

Question 12

Parallel Lines

Answer 12

Parallel lines have the same slope and different y-intercept. The lines will never intercept.

Question 13

Reflection: Points

Answer 13

• Reflection is denoted by the use of “prime” notation: point P is reflected → P’
• Reflection over the x-axis: (x, y) → (x, -y)
• Reflection over the y-axis: (x, y) → (-x, y)
• Reflection over the origin: (x, y) → (-x, -y)
• **Reflection over a point (a,b) → (a,b) is the midpoint. Use midpoint formula.
• **Reflection over y = x: (x, y) → (y, x)
• **Reflection over y = -x: (x, y) → (-y, -x)
• **Reflection over y = b: (x, y) → (x, 2b - y)
• **Reflection over x = a: (x, y) → (2a - x, y)

Concept can be applied to lines (?) → If a line y = 2x + 3 is reflected over y = x, the image line is x = 2y + 3

y = x → THINK: Just swapping the two, so (x, y) → (y, x)

Question 14

Reflection: Line Segments

Answer 14

To reflect a line segment over the x-axis, y-axis, or origin, reflect the endpoints of the line segment. Once we have the two new endpoints, we can draw a line segment connecting them, and this new line segment is the reflection of the original.

Question 15

Reflection: Polygon

Answer 15

To reflect a polygon over the x-axis, the y-axis, or the origin, we can reflect each vertex of the polygon, and then connect the reflected vertices to form the reflected polygon.

Question 16

Distance between Two Points

Answer 16

• General formula: √(x₂ - x₁)² + (y₂ - y₁)²
• Two points with the same y-coordinate: Distance = absolute value of the difference between their x-coordinates
• Two points with the same x-coordinate: Distance = absolute value of the difference between their y-coordinates

Question 17

Midpoint of a Line

Answer 17

Midpoint(x_m, y_m) = ( (x₁ + x_{<span>2</span>}) / 2, (y_{<span>1</span>} + y₂) /2 )

x_m = x-coordinate of the midpoint

y_m = y-coordinate of the midpoint

Question 18

Equation of a Circle on the Coordinate Plane

Answer 18

The equation of a circle centered at the point (a, b) is

(x - a)² + (y - b)² = r² or (a - x)² + (b - y)² = r²

(Concept: basically pythagorean theorem)

So when the center is at the origin, i.e., (a, b) = (0, 0), the equation is x² + y² = r²

3 points uniquely define a circle

Question 19

Graphing Inequalities

Answer 19

All the inequalities need to be in the slope-intercept form

• “Greater than” means y > mx + b
• “Less than” means y < mx + b

Question 20

2 Lines Intersecting

Answer 20

• Never intersect: Same slope but different y-intercept (parallel lines)
• Always intersect: Same slope and same y-intercept (same line)
• Intersect at some point: Different slopes (not parallel lines)

Question 21

2 Circles Intersecting

Answer 21

• To determine whether two circles intersect each other, knowing the respective equations of the two circles is enough to solve the question. Also, don’t actually solve the system of equations in a DS question.
• Don’t mistakenly conclude that two circles must intersect simply because the center of one circle is contained within the other circle (e.g., one circle can be so small that it is contained entirely within the larger circle without ever intersecting the larger circle).

Question 22

For a point on an xy-coordinate plane to be equidistant to two points…

Answer 22

the point needs to lie on the perpendicular bisector of the line segment with those endpoints