Mathematics notes for standardized tests-2


Formulas and properties

Geometry:

If two lines (not parallel) cut by traversal, only vertical angles are equal.
If two parallel lines  cut by traversal, consecutive interior angles will be supplementary.

Sum of exterior angles in any polygon is $360^o$
Sum of exterior angles in polygon with $n$ sides is $(n-2)\times 180^o$.

For 30-60-90 right angle triangle-

Hypotenuse is twice  the length of the side opposite to $30^o$ and other leg is $\sqrt{3}$ times. Here $c=2a$ and $b=a\sqrt{3}$.

For 45-45-90 right angled triangle-
Two legs have same length and Hypotenuse is $\sqrt{2}$ times. Here $c=\sqrt{2}a$

For similar triangle-

$\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}$ and $\frac{BC}{EF}=\frac{AX}{DY}$



Circle:
 
When two chords intersects in a circle product of segment on one chords is equal to the product of segment in other chords.
Here $AE\times EB = CE\times ED$.

Degree measured of inscribed angle $\angle CEB=\frac{1}{2}(arc BC+ arc AD)$

Polygon Perimeter Area
Triangle a+b+c $\frac{1}{2}bh$
Trapezoid a+b+c+d $\frac{1}{2}\times h \times (a+b)$ h=distance between parallel sides
Parallelogram a+b+c+d bh
Rectangle 2l+2w lw
Square 4a $a^2$
Circle $2\pi r$ $\pi r^2$


For solids:


Solid Surface area Volume
Rectangular solid 2lw+2lh+2wh lwh
Cube $6a^2$ $a^3$
Right circular cylinder $2\pi r h +2\pi r^2$ $\pi r^2 h$
Sphere $4\pi r^2$ $\frac{4}{3}\pi r^3$
Right circular cone $\frac{1}{3}\pi r^2 h$ $\pi r^2+\pi r s$
Pyramid - $\frac{1}{3}\times B \times h$ B= area of base

For two points$P=(x_1,y_2)$ and $Q=(x_2,y_2)$ the distance between them
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Mid point=$\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$

Slope=$\frac{y_2-y_1}{x_2-x_1}$

If slopes $m_1=m_2$, lines are parallel
If $m_1\times m_2=-1$ lines are perpendicular

At x intercept, $y=0$
At y intercept, $x=0$