1.3 Coordinate Geometry in the (x, y) Plane

• What is coordinate geometry in the (x, y) plane the study of?

  Geometric shapes using coordinates
• The x-axis is the horizontal line, and the y-axis is the vertical line, intersecting at the origin
• Quadrant I includes points with $x > 0$ and $y > 0$
• What are the signs of the coordinates in Quadrant II?

  $x < 0$ and $y > 0$
• Match the quadrant with its coordinate signs:

  Quadrant I ↔️ $x > 0$, $y > 0$

  Quadrant II ↔️ $x < 0$, $y > 0$

  Quadrant III ↔️ $x < 0$, $y < 0$

  Quadrant IV ↔️ $x > 0$, $y < 0$
• To plot a point, the x-coordinate indicates how far to move left or right from the origin
• A negative y-coordinate means you move upward from the origin.

  False
• Steps to plot a point on the coordinate plane:

  1️⃣ Start at the origin $(0, 0)$

  2️⃣ Move along the x-axis to the x-coordinate

  3️⃣ Move along the y-axis to the y-coordinate

  4️⃣ Mark the point
• What is the distance formula for two points P_{1}(x_{1}, y_{1})</latex> and $P_{2}(x_{2}, y_{2})$?


  $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
• The distance between the points $(2, 3)$ and $(6, 7)$ is $4\sqrt{2}$
• Match the step in calculating distance with its action:

  Identify coordinates ↔️ $x_{1}, y_{1}, x_{2}, y_{2}$

  Apply distance formula ↔️ $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

  Simplify ↔️ $d =$$\sqrt{32} =$$4\sqrt{2}$
• The x-axis and y-axis intersect at the origin $(0, 0)$.
• What does the y-coordinate of a point indicate?

  Vertical movement from origin
• What does the x-coordinate indicate when plotting a point on the coordinate plane?

  Left or Right
• The x-coordinate indicates movement left (-) or right
• The first step to plot a point is to start at the origin $(0, 0)$.
• Steps to plot a point on the coordinate plane

  1️⃣ Start at the origin $(0, 0)$

  2️⃣ Move along the x-axis to the x-coordinate

  3️⃣ Move along the y-axis to the y-coordinate

  4️⃣ Mark the point
• What formula is used to calculate the distance between two points in the coordinate plane?

  $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
• The distance formula is given by d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}</latex>
• To calculate the distance between two points, you first identify the coordinates and then apply the distance formula.
• What is the definition of the midpoint of a line segment?

  Halfway between endpoints
• The midpoint formula is Midpoint = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right)</latex>
• To find the midpoint, you average the x-coordinates and the y-coordinates of the endpoints.
• What are the three common forms for the equation of a straight line?

  Slope-intercept, point-slope, general
• The slope-intercept form of a line is $y =$$mx +$$c$
• If given two points, the first step to find the equation of the line is to calculate the slope.
• What is coordinate geometry in the (x, y) plane used to study?

  Geometric shapes using coordinates
• What formula is used to calculate the slope of a line?

  $m =$$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
• The point-slope form is used with one of the given points
• The point-slope form is used directly when given a point and slope.
• What is the equation of the line passing through $(2, 3)$ with slope $m =$$2$ in slope-intercept form?


  $y =$$2x - 1$
• Steps to find the equation of a line in different forms

  1️⃣ Calculate the slope

  2️⃣ Use point-slope form

  3️⃣ Simplify to slope-intercept form

  4️⃣ Convert to general form
• What does coordinate geometry study in the (x, y) plane?

  Geometric shapes
• The x-axis and y-axis intersect at the origin
• What are the conditions for a point to lie in Quadrant I?

  x > 0, y > 0</latex>
• Match the quadrant with its coordinate conditions:

  Quadrant II ↔️ $x < 0, y > 0$

  Quadrant III ↔️ $x < 0, y < 0$

  Quadrant IV ↔️ $x > 0, y < 0$
• What does the y-coordinate indicate when plotting a point?

  Up or down movement
• The first step to plot a point is to start at the origin
• To plot $(3, - 2)$, you move 3 units right and 2 units down from the origin.
• What is the formula used to calculate the distance between two points in the coordinate plane?

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