Geometry Flashcards
When n lines intersect through a common point, the sum of the angles created by the those n lines is…
360
Angles are supplementary if..
Angles are supplementary if their measures add up to 180°.
For triangles, the longest side of the triangle is always opposite…
the largest angle
For triangles, the shortest side of the triangle is always opposite…
the smallest angle
Sum of several exterior ADJACENT angles of a polygon must equal…
NOT mirroring angles
360
taking one exterior angle at each vertex. if more than one, treat each additional as another part of a 360 add up
An exterior angle of a triangle is equal to the sum of the two remote interior angles
‘remote’ angles are those that are not immediately next to exterior angles
Sum of lengths of any two sides of a triangle must…
Be greater than the length of the third size
A + B > C
B + C > A
A + C > B
Difference of lengths of any two sides of a triangle must…
Be less than the length of the third size
A - B < C
B - C < A
A - C < B
Acute angle is
less than 90 degrees
obtuse angle is
greater than 90 degrees but less than 180
Two most common pythagorean triples
3-4-5 [incl any multiples such as 6,8,10 and 9,12,15]
5-12-13 [incl any multiples as as 10,24,26]
what are the angles of an isosceles right triangle
45-45-90
2 legs are equal
area = leg^2/2
Ratio of sides of 45-45-90 right triangle [iscoeles]
x : x : x√2
where x is the length of the two equal legs, x√2 is hypotenuse
if we know the size of one leg, we can determine the size of all legs
Ratio of sides of 30-60-90 right triangle
x : x√3 : 2x
x = side opposite of 30 degree angle
x√3 = side opposite of 60 degree angle
2x = side opposite of 90 degree angle
if we know the size of one leg, we can determine the size of all legs
Area for equilateral triangle
Area=s^2√3/4
where s is one size
the area can only be determined only when the height is known
What are quadrilaterals?
four sided polygons, including squares, rectangles, parallelograms, rhombuses, and trapezoids
all squares are rectangles, all rectangles are parallelograms
what is the equation for the diagonal of any rectangle?
√l^2+w^2
Properties of a square
- 4 right angles
- 4 sides of equal length
- each diagonal bisects the square into 2 45-45-90 triangles
- length of square’s diagonal [hypotenuse] = s√2 where s is the length of one side of square
Given a rectangle with a fixed perimeter, the rectangle with the maximum area is a square.
Given a rectangle with a fixed perimeter, the rectangle with the maximum area is a square.
Given a rectangle with a fixed area, the rectangle with the smallest perimeter is a square.
Given a rectangle with a fixed area, the rectangle with the smallest perimeter is a square.
Trapezoid
Trapezoid is a quadrilateral where the bases are parallel but the other pair of opposite sides are not parallel
area of a trapezoid
(base 1 + base 2) * height / 2
sum of interior angles of a polygon
sum of interior angles of polygon = (n-2) x 180
where n = number of sides
measure of any one interior angle in a polygon
180(n-2)/n
Hexagon
- six equal sides and six equal angles
- interior angles add up to 720 degrees
- each interior angle measures to 120 degrees
Area of a hexagon
3√3/2 * (s^2) where s is length of one of the sides
How to label the quadrants of the XY plane
start in top right corner with I and go counter-clockwise with 2, 3, 4
Slope of a line formula
y2 - y1 / x2 - x1
Slope of a line
The slope of a line can be
- zero (a horizontal line)
- undefined (a vertical line)
- positive (a line that rises when the line moves from left to right)
- or negative (a line that falls when the line moves from left to right).
what is an intercept?
Point where line crosses the x and y axises
Lines with positive slopes - golden rule
will always intersect quadrants I and III
Lines with positive slopes - x intercept is negative
y intercept is positive, will intersect quadrants I, II, III
Lines with positive slopes - x intercept is zero
will only intersect quadrants I and III
Lines with positive slopes - x intercept is positive
y intercept is negative, will intersect quadrants I, III, IV
3 equivalent circle ratios
central angle / 360 = arc length/circumference = area of sector / area of circle
Angles inscribed in a circle
The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts.
The legs of an isoceles triangle can be the radii of a circle
if legs start at beginning of circle and end at circumfrence, it is an isoceles triangle with equal legs 45-45-90
If an equilateral triangle is inscribed in a circle, the center of the triangle is the center of the circle
a line drawn from the center of the triangle to a vertex of the base creates a 30-60-90 triangle. the line would also create a radius
What is midpoint formula for a coordinate graph
X1+X2/2 and Y1+Y2/2
basically finding average of both points
Distance between two points formula
d=√((x2 – x1)² + (y2 – y1)²).
y = mx + b
y = y point of (x,y) pair
m = slope = y2-y1/x2-x1 or rise over run
x = x point of (x,y) pair
b = where the line crsses the y intercept [the vertical line]
the slopes of two perpendicular lines will always multiply together to equal
negative 1
if two lines are parallel, they have…
same slope
