1.3 Coordinate Geometry in the (x, y) Plane

• The origin in the Cartesian coordinate system is denoted as (0, 0).


  True
• Steps to plot a point in the (x, y) plane

  1️⃣ The x-coordinate represents the horizontal distance from the origin

  2️⃣ The y-coordinate represents the vertical distance from the origin

  3️⃣ The point is plotted at the intersection of the horizontal and vertical distances
• The midpoint formula calculates the coordinates of the point that lies exactly halfway between two given points.
• What is the midpoint formula used for in the Cartesian coordinate system?

  Calculating the midpoint
• What is the x-coordinate of the midpoint between (2, 3) and (6, 7)?

  4
• What are the two perpendicular axes in the Cartesian coordinate system called?

  x-axis and y-axis
• In the (x, y) plane, what does the y-coordinate represent?

  Vertical distance
• What is the formula for the distance between two points with the same y-coordinate?

  $d =$$|x_{2} - x_{1}|$
• The slope-intercept form of a straight line is $y =$$mx +$$b$
• What is the slope-intercept form of the equation of a straight line?

  $y =$$mx +$$b$
• The slope of a line is calculated as the rate of change between two points on the line.


  True
• Parallel lines have the same slope.
• What is the relationship between the slopes of parallel lines?

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

  X-axis and y-axis
• The x-coordinate represents the horizontal distance from the origin.


  True
• The distance formula is derived from the Pythagorean theorem.


  True
• What is the slope-intercept form of the equation of a straight line?

  $y =$$mx +$$b$
• What is the slope condition for parallel lines?

  $m_{1} =$$m_{2}$
• The perpendicular distance from (3, 4) to $2x - y +$$5 =$$0$ is approximately 3.13$units$
• The formula for calculating the perpendicular distance from a point to a line is \frac{|Ax_{1} +$By_{1} +$ C|}{\sqrt{A^{2} + B^{2}}}
• What is the shortest distance from a point to a line called?

  Perpendicular distance
• The intersection point of two lines is the point where they meet
• Substituting the x-coordinate back into either original equation gives the y-coordinate of the intersection point.


  True
• What does $(a, b)$ represent in the standard equation of a circle?


  Center of the circle
• What does the x-coordinate of a point represent in the Cartesian coordinate system?

  Horizontal distance from origin
• Each point in the plane is uniquely identified by an ordered pair of coordinates.


  True
• The origin is located at (0, 0).


  True
• Steps to plot a point in the (x, y) plane

  1️⃣ The x-coordinate represents the horizontal distance from the origin.

  2️⃣ The y-coordinate represents the vertical distance from the origin.

  3️⃣ The point is plotted at the intersection of the horizontal and vertical distances.
• The formula for horizontal distance is $d =$$|x_{2} - x_{1}|$ when the y-coordinates are the same
• The midpoint between (2, 3) and (6, 7) is (4, 5).

  True
• How is the y-intercept $b$ found in the slope-intercept form?


  $b =$$y - mx$
• The slope-intercept form of a line equation is $y =$$mx +$$b$, where $m$ is the slope and $b$ is the y-intercept
• Steps to find the equation of a line given two points (2, 3) and (5, 7)

  1️⃣ Calculate the slope (m): $m =$$\frac{7 - 3}{5 - 2} =$$\frac{4}{3}$

  2️⃣ Find the y-intercept (b): $b =$$3 - \frac{4}{3} \times 2 =$$\frac{1}{3}$

  3️⃣ Write the equation in slope-intercept form: $y =$$\frac{4}{3}x +$$\frac{1}{3}$
• Match the linear equation form with its key components:

  Slope-intercept ↔️ Slope (m), Y-intercept (b)

  Point-slope ↔️ Slope (m), Point $(x_{1}, y_{1})$

  Standard ↔️ Coefficients (A, B, C)
• The Cartesian coordinate system uses two perpendicular axes: the horizontal x-axis and the vertical y-axis.
• What does the y-coordinate represent when plotting a point in the (x, y) plane?

  Vertical distance
• What is the formula to find the distance between two points (x1, y1) and (x2, y2)?

  $d =$ \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}
• The diagonal distance formula is the same as the general distance formula.


  True
• Match the component with its formula in the midpoint formula:

  Midpoint x-coordinate ↔️ \frac{x_{1} + x_{2}}{2}

  Midpoint y-coordinate ↔️ \frac{y_{1} + y_{2}}{2}
• The formula for the x-coordinate of the midpoint is \frac{x_{1} + x_{2}}{2}