The height/altitude of a triangle is always perpendicular to the...
base
The longest side of a triangle is always the side...
opposite the greatest angle
The shortest side of a triangle is always the side...
opposite the smallest angle
A right angle triangle is defined by what? And what calculation allows you to calculate the length of one side if you know the other two sides?
Right angle triangle = has one angle equal to 90 deg
Pythagorean Theorem
a^{2} + b^{2} = c^{2}
What are the two most common values (or ratio of) used on the GMAT for right triangles & their side lengths?
3,4,5 or 5,12,13
Two other very common right triangles are?
30/60/90
& 45/45/90 (AKA Right Isosceles Triangle)
30/60/90 Triangles always have sides in what proportions?
1, √3, 2 or x, x√3, 2x
For any right triangle you need how many sides to solve for another side?
Need 2 sides to solve for 1
In a 30/60/90 triangle 2x = short, med, or long side? And where is located?
2x = longest side & is always opposite the 90 degree (on the hypotenuse)
In a 30/60/90 triangle x = short, med, or long side? And where is located?
x (or 1) = short. Always opposite the 30 deg. angle
In a 30/60/90 triangle x√3 = short, med, or long side? And where is located?
x√3 = med side. Always opposite the 60° angle
a 45/45/90 triangle always has sides with lengths of the proportion...
1, 1, and √2 or x, x, x√2
What's the definition of an Isosceles Triangle?
Has at least two sides and two angles that are the same.
What in an isosceles triangle creates two equal triangles? And how?
The altitude splits the base & the angle evenly when drawn from the unequal side. Creating two congruen (equal) triangles on each side.
What's the definition of an Equilateral triangle? What's it also considered?
Definition: All sides are equal length & all angles are 60°. Also is considered an Isosceles triangle.
When you take the height of an equilateral triangle (split in two) you form what?
two 30/60/90 triangles.
What is the height of an equilateral triangle equal to?
√3 times the side of the triangle
2
Area of an equilateral triangle?
(s^{2}√3)
4
s = sides
If you have an equilateral triangle split into a 30/60/90 triangle what are the proportions?
30° side (x) = s/2
60° side (x√3) = (s√3)/2
90° side (2x) = s
s = side
Whenever a supplementary angle is drawn off of an interior angle of a triangle - it will always equal the sum of the other two interior angles of the triangle.
c° + d° = 180 a°+b°+c°=180 thus: a°+b° = d°
What's a similar triangle?
All the angles are the same and the proportion of the corresponding sides & heights are constant.
Little Triangle Sides = a,b,c
Large Triangle Sides = 2a, 2b, 2c (for ex)
What are the 3 standard ways to identify triangles that are similar?
1) AAA (angle, angle, angle) if 2 or 3 angles match b/w triangles, they are similar.
2) SSS (side, side, side) if there is a constant ratio of corresponding sides, they are similar.
3) SAS (side, angle, side) if there is a constant ratio of corresponding sides for 2 sets of sides AND the angle between those two sides is the same as the other triangle, they are similar.
What is the property of any triangle involving the lengths of its sides?
The sum of any two lengths is greater than the length of the third side.
What's the relationship b/w 2 sides of a triangle vs. the third side?
The sum of the two smaller sides has to be greater than the third side.