#NumberAndAlgebra #Path #Adv >[!info]- [Linear relationships C (Path) | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fab720e184) >- MA5-LIN-P-01 describes and applies transformations, the midpoint, gradient/slope and distance formulas, and equations of lines to solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | -------------------------- | -------------------------------------------------------- | -------------------------------------------------------------------------------------- | | [[Linear Relationships B]] | - midpoint, gradient, distance | - midpoint, gradient, distance formulas | | [[Equations C]] | - rearranging equations<br>- solving multistep equations | - rearranging equation of a line<br>- finding intercepts, using point-gradient formula | ## [[Apply formulas to find the midpoint and gradient slope of an interval on the Cartesian plane.pdf]] - Apply the formula to find the midpoint of the interval joining 2 points on the Cartesian plane: $M\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ - Use the relationship $m=\frac{\text{rise}}{\text{run}}$ to establish the formula for the gradient/slope $m$ of the interval joining the 2 points $(x_1,y_1)$ and $(x_2,y_2)$ on the Cartesian plane: $m=\frac{y_2-y_1}{x_2-x_1}$ - Apply the gradient formula to find the gradient of the interval joining 2 points on the Cartesian plane ## [[Apply the distance formula to find the distance between 2 points located on the Cartesian plane.pdf]] - Apply knowledge of Pythagoras’ theorem to establish the formula for the distance $d$ between the 2 points $(x_1,y_1)$ and $(x_2,y_2)$ on the Cartesian plane: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$ - Apply the distance formula to find the distance between 2 points on the Cartesian plane ## [[Use various forms of the equation of a straight line.pdf]] - Rearrange linear equations from gradient–intercept form $y=mx+c$ to general form $ax+by+c$ and vice versa - Find the $x$- and $y$-intercepts of a straight line in any form - Graph the equation of a straight line in any form - Use the point–gradient form $y-y_1=m(x-x_1)$ or the gradient–intercept form $y=mx+c$ to find the equation of a line passing through a point $(x_1,y_1)$, with a given gradient $m$ - Use the gradient and the point–gradient form to find the equation of a line passing through 2 points - Find the equation of a line that is parallel or perpendicular to a given line in any form - Determine and justify whether 2 given lines are parallel or perpendicular ## [[Solve problems by applying coordinate geometry formulas.pdf]] - Solve problems including those involving geometrical figures by applying coordinate geometry formulas ## [[Identify line and rotational symmetries.pdf]] - Identify lines (axes) and rotational symmetry in plane shapes - Identify line and rotational symmetry in various linear and non-linear graphs ## [[Describe translations, reflections in an axis, and rotations through multiples of 90 degrees on the Cartesian plane, using coordinates.pdf]] - Apply the notation $P'$ to name the image resulting from applying a transformation to a point $P$ on the Cartesian plane - Determine and plot the coordinates for $P'$ resulting from translating $P$ one or more times - Determine and plot the coordinates for $P'$ resulting from reflecting $P$ in either the $x$- or $y$-axis - Determine and plot the coordinates for $P'$ resulting from rotating $P$ by a multiple of 90° about the origin