Mathematical Formulas - Secondary 4

||y=ax+b|||a|: rate of change (slope)

|b|: y-intercept||a=\dfrac{y_2-y_1}{x_2-x_1}||

||\dfrac{x}{a}+\dfrac{y}{b}=1|||a|: x-intercept

|b|: y-intercept

|\Rightarrow| General

Find the common denominator and bring everything to the same side of the equation.

|\Rightarrow| General

Find the common denominator and bring everything to the same side of the equation.

Number of zeros

Directly accessible from the equation (see the box above).

Note: if there are no zeros, it's not possible to use this form.

Value of the zeros

|z_1| and |z_2|

|A =\dfrac{b\times h}{2}|

|A = \sqrt{p(p-a)(p-b)(p-c)}|
where
|p=\dfrac{a+b+c}{2}=| half-perimeter

|A=\dfrac{ab\sin C}{2}|
where |C=| measure of the angle located between sides |a| and |b|

• Pythagorean theorem
  ​In any right triangle, the sum of the square of the legs |(a| and |b)| is equal to the square of the hypotenuse |(c).|||a^2+b^2 = c^2||

• In any triangle, the measure of any one side is smaller than the sum of the measures of the other two sides.

• In any isosceles triangle, the angles opposite the congruent sides are congruent.

• In any right triangle, the acute angles are complementary |(90^\circ).|

• Any right triangle with a |30^\circ| angle has a side opposite the |30^\circ| angle that is equal to half the hypotenuse.

Altitude to Hypotenuse theorem

In a right triangle, the height |(h)| drawn from the right angle is the proportional mean between the 2 segments it creates on the hypotenuse |(m| and |n).|||\dfrac{m}{h}=\dfrac{h}{n}\quad\text{or}\quad h^2=mn||

Product of the Sides theorem

In a right triangle, the product of the hypotenuse |(c)| and the corresponding height |(h)| is equal to the product of legs |(a| and |b).|||ch=ab\quad\text{or}\quad h=\dfrac{ab}{c}||

Proportional Mean theorem

In a right triangle, each leg |(a| and |b)| is the proportional mean between its projection onto hypotenuse |(m| or |n)| and the entire hypotenuse |(c).|||\dfrac{m}{a}=\dfrac{a}{c}\quad\text{or}\quad a^2=mc\\\dfrac{n}{c}=\dfrac{b}{c}\quad\text{or}\quad b^2=nc||

||t_{(a,b)}:(x,y)\stackrel{t}{\mapsto}(x+a,y+b)||

​||t^{-1}_{(a,b)}=t_{(-a,-b)}:(x,y)\stackrel{t}{\mapsto}(x-a,y-b)||

||\begin{align}r_{(O,90^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-y,x)\\[3pt]r_{(O,-270^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-y,x)\\[3pt]r_{(O,180^\circ)}&:(x,y)\stackrel{r}{\mapsto}(-x,-y)\\[3pt]r_{(O,-90^\circ)}&:(x,y)\stackrel{r}{\mapsto}(y,-x)\\[3pt]r_{(O,270^\circ)}&:(x,y)\stackrel{r}{\mapsto}(y,-x)\end{align}||

||\begin{align}​r^{-1}_{(O,90^\circ)}&=r_{(O,-90^\circ)}\\[3pt]r^{-1}_{(O,-270^\circ)}&=r_{(O,270^\circ)}\\[3pt]r^{-1}_{(O,180^\circ)}&=r_{(O,180^\circ)}\\[3pt]r^{-1}_{(O,-90^\circ)}&=r_{(O,90^\circ)}\\[3pt]r^{-1}_{(O,270^\circ)}&=r_{(O,-270^\circ)}\end{align}||

​Reflection

(Symmetry)

||\begin{align}​s_x&:(x,y)\stackrel{s}{\mapsto}(x,-y)\\[3pt]s_y&:(x,y)\stackrel{s}{\mapsto}(-x,y)\\[3pt]s_{\small/}&:(x,y)\stackrel{s}{\mapsto}(y,x)\\[3pt]s_{\tiny\backslash}&:(x,y)\stackrel{s}{\mapsto}(-y,-x)\end{align}||

||\begin{align}​s^{-1}_x&=s_x\\[3pt]s^{-1}_y&=s_y\\[3pt]s^{-1}_{\small/}&=s_{\small/}\\[3pt]s^{-1}_{\tiny\backslash}&=s_{\tiny\backslash}\end{align}||

||h_{(O,k)}:(x,y)\stackrel{h}{\mapsto}(kx,ky)||

​||h^{-1}_{(O,k)}=h_{\left(\frac{1}{k},\frac{1}{k}\right)}:(x,y)\stackrel{h}{\mapsto}\left(\dfrac{x}{k},\dfrac{y}{k}\right)||

||R=\text{Boundary}_\text{upper}-\text{Boundary}_\text{lower}||

||r\approx\pm\left(1-\dfrac{l}{L}\right)|| where |L| represents the length and |l,| the width of the rectangle that encompasses the scatter plot.

The sign of |r| depends on the direction of the scatter plot.