Precalc +: LOS, LOC, Area

Triangle Congruence Cases

SSS, SAS, ASA, AAS, SSA/HL

Triangle Similarity Case

AAA

Law of Sines

sin(A)/a = sin(B)/b = sin(C)/c \[ASA/AAS\]

***NEVER USE TO FIND A POTENTIALLY OBTUSE ANGLE***

Law of Cosines

a² = b² + c² - 2bc(cos(A))

b² = a² + c² - 2ac(cos(B))

c² = a² + b² - 2ab(cos(C))

\[SSS/SAS\]

Cases when there are no triangles

* Sum of the 2 smaller sides is less than or equal to largest side
* Angle is over 180°
* Biggest angle does not correspond to longest side

Special Triangle Congruence Case

SSA/HL (0, 1, or 2 triangles)

\# of triangles when Opp=Adj (given obtuse angle)

0

\# of triangles when Opp

0

\# of triangles when Opp>Adj (given obtuse angle)

1

\# of triangles when Opp=h (given acute angle)

1

\# of triangles when Opp

0

\# of triangles when Opp=Adj (given acute angle)

1

\# of triangles when Opp>Adj (given acute angle)

1

\# of triangles when h

2

How to draw Special Case

* Draw a start line (horizontal)
* Set up the fixed angle on the left side of the line
* Label the line connected to the angle the adjacent side
* Connect a line to the end of the adjacent side and call it the opposite/swing line (*DON’T CONNECT LINE*)
* If there are two triangles, set up both cases.

Height of an SSA triangle

sin(given angle) = h/adj

\[*ONLY* find h is given angle is acute and opp

Area of a triangle

A= (1/2)bh

Area of a SAS triangle

A = (1/2)ab(sin(C))

A = (1/2)bc(sin(A))

A = (1/2)ac(sin(B))

Area of a SSS Triangle (Heron’s)

A = √s(s-a)(s-b)(s-c)

s = (1/2)(a+b+c)

Area of ASA or AAS triangle

Use LOS/LOC to find a piece then calculate area

Properties of a regular pentagon

* Sides and angles are congruent
* 5 Radii form 5 isocoles triangles
* Apothems bisect the sides and angles created by the triangles around the center

Properties of a regular hexagon

* 6 Radii form Equilateral triangles
* Radius=Side length
* Apothems bisect the sides and the angles created by the triangles around the center
* Apothems form 30-60-90 triangles

Properties of an inscribed pentagon

* Radius of pentagon=radius of circle

Properties of an inscribed circle

* Apothem=radius of circle