3.4 C: Coordinate Geometry in the (x, y) Plane

• What are the two perpendicular axes in the Cartesian coordinate system called?

  x-axis and y-axis
• Match the quadrant with the sign of its x and y coordinates:

  Quadrant I ↔️ Positive, Positive

  Quadrant II ↔️ Negative, Positive

  Quadrant III ↔️ Negative, Negative

  Quadrant IV ↔️ Positive, Negative
• The point (-2, 5) lies in Quadrant II.


  True
• Match the quadrant with the signs of its x and y coordinates:

  Quadrant I ↔️ (+, +)

  Quadrant II ↔️ (-, +)

  Quadrant III ↔️ (-, -)

  Quadrant IV ↔️ (+, -)
• The coordinates of a point are given as an ordered pair (x, y), where x represents the horizontal position.
• The distance formula is derived from the Pythagorean theorem.
• What is the distance between points A(2, 3) and B(5, 7)?

  5 units
• The midpoint of a line segment with endpoints (2, 3) and (6, 9) is (4, 6).


  True
• The midpoint of a line segment with endpoints (2, 3) and (6, 9) is (4, 6).


  True
• In which quadrant does the point (3, -2) lie?

  IV
• What is the distance formula in coordinate geometry?

  $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
• What theorem is the distance formula derived from?

  Pythagorean Theorem
• Match the formula with its description:

  Distance Formula ↔️ $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

  Pythagorean Theorem ↔️ $a^{2} +$$b^{2} =$$c^{2}$
• What is the formula for finding the midpoint of a line segment with endpoints (x_{1}, y_{1})</latex> and $(x_{2}, y_{2})$?


  $M =$ \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\right)
• Match the gradient type with its description:

  Positive ↔️ Increases upwards from left to right

  Negative ↔️ Decreases downwards from left to right

  Zero ↔️ Flat and horizontal

  Undefined ↔️ Vertical
• In the slope-intercept form $y =$$mx +$$b$, what does 'm' represent?


  Slope
• What is the direction of a line with an undefined gradient?

  Vertical
• The slope of a line represents its steepness and direction
• Match the given information with the steps to find m and b:

  Two points on the line ↔️ Calculate the slope and substitute into $y =$$mx +$$b$

  Slope and one point on the line ↔️ Substitute the slope and point into $y =$$mx +$$b$

  Slope and y-intercept ↔️ Use the given slope and y-intercept values
• What is the first method to analyze the intersection of two lines?

  Graphing
• In the substitution method, you solve one equation for either x or y
• What is the name of the point where the x-axis and y-axis intersect in the Cartesian coordinate system?

  Origin
• The point (3, -2) lies in Quadrant IV


  True
• The coordinate plane in the Cartesian coordinate system is divided into four quadrants.


  True
• To plot a point in the Cartesian coordinate system, you need to identify its coordinates
• Steps to find the distance between two points in a Cartesian plane using the distance formula

  1️⃣ Identify the coordinates of the two points

  2️⃣ Substitute the coordinates into the distance formula

  3️⃣ Calculate the squared differences

  4️⃣ Find the square root of the sum
• What is the distance between the points A(2, 3) and B(5, 7)?

  5 units
• Steps to find the midpoint of a line segment

  1️⃣ Identify the coordinates of the endpoints

  2️⃣ Calculate the average of the x-coordinates

  3️⃣ Calculate the average of the y-coordinates

  4️⃣ Combine the averages into a coordinate pair
• A line with a positive gradient increases from left to right.


  True
• What is the equation of the line passing through points (2, 5) and (4, 9)?

  $y =$$2x +$$1$
• The slope-intercept form of a line is y = mx + b.
• To calculate the slope between two points, use the formula m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$.
• The slope of the line passing through (2, 5) and (4, 9) is 2.

  True
• The coordinate plane is divided into four quadrants
• In which quadrant would the point (3, 4) be plotted?

  Quadrant I
• The coordinate plane is divided into four quadrants
• What are the signs of the x and y coordinates in Quadrant I?

  Positive and positive
• In which quadrant would you plot the point (3, 4)?

  Quadrant I
• The Pythagorean theorem is $a^{2} +$$b^{2} =$$c^{2}$.


  True
• What is the midpoint formula for a line segment with endpoints $(x_{1}, y_{1})$ and (x_{2}, y_{2})</latex>?


  $M =$ \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\right)