regents geometry mc and review Flashcards
finding area/perimeter for triangles
1) use perimeter/area of whatever particular shape ex. A = s2 and P= 4s for a square, for a parallelogram/rhombus/rectangle Area= bh Perimeter= 2b+ 2h
2) use distance formula (for triangles, use it for the sides that isn’t the base or height)
3)
reflecting a shape over a particular line
ex. triangle ABC, with vertices A (0,0) B (3,5), C (0,5) is graphed, what is its height and radius length when reflected?
be wary of which line of the shape is being reflected and take into consideration how the new perspectiv of the 3-d shape changes the height/radius length (they switch often with the way the shape faces)
centroid
2/3rds distance from each vertex from their midpoints from each side,
point of concurrency for the medians
How to find altitude (slope of line, flipped and negated, with vertex of triangle)
1) find slope of the side the altitude intersects and flip & negate it
2) Use the vertex (where altitude begins) and the slope to set up an equation
How to find median (random pair of coordinates from line with slope, made up from vertex & midpoint of that line)
1) find MIDPOINT!!!! of the line the median intersects
2) take those coordinates + the vertex’s coordinates to find slope of the median
3) use THIS SLOPE and a RANDOM pair of coordinates from the line to set up the equation
proportion problems
BE CAREFUL of where the numbers are, make sure your proportion ise evenly set up
reflections
positive reflection is clockwise, negative reflection is counterclockwise
a rigid motion is a transformation in which the pre image and image are
congruent
the rigid motion transformations (preserving distance and angle measure)
reflection, rotation, translation
reflection
image and pre image will have opposite orientations, they will have different orientations with an odd amount of reflections/single flip and the same orientations with an even #
reflection over x axis (x,y)
(x,-y)
reflection over y axis (x,y)
(-x, y)
reflected over y= x (x,y)
(-y, x)
reflected over -y=-x
(-y,-x)
positive rotation
clockwise
negative rotation
counterclockwise
when applying transformations
apply the 2nd one first
glide reflection
reflection + translation
invariant point
image has the same coordinates as the original point under a transformation
congruency transformation
pre image and image are congruent
similarity transformation
pre image and image are similar
dilation
1) plug in equation on calculator and search for center point that the line is supposed to be dilated by
2) if it is there, nothing changes. if it isn’t, it’s parallel so the slope doesn’t change but the y-intercept does.
how to tell if a shape carries onto itself
envision it/draw it and preform the transformation to see if the shape looks the same
Rectangle A’ B’ C’ D’ is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3rds so
perimeter is in the same ratio as the sides, area is the square of the ratio of its sides so A’ B’ C’ D’ has the perimeter that is 2/3rds the perimeter of rectangle ABCD
Area of a triangle on coordinate geometry
0.5 (1/2) (x1 (y2-y3) + x2 (y3-y1) + x3 (y1-y2)
altitude
works as height when solving for area of a triangle on a coordinate
area of a sector
x/360 pi (r) 2
area of arc length
x/360 x 2(pi)(r)
median
goes to midpoint of the opposite side (perpendicular bisectors technically do this too which is why they classify as such and the triangles wind up isosceles with reflexive and 1/2 of each base
if a composition of trnasformations includes a dilation
the transformation is similar
a composition of line reflections over 2 parallel lines (or any even # of II lines) are equal to
a translation
a composition of line reflections over 3 II lines (or any odd # of II lines) is equivalent to
a single reflection
how to rotate/dilate a figure about a center than the origin
1) translate the center of rotation to the origin
2) apply that translation to the points of the actual figure and then translate back
translations/dilations of circles can prove that
all circles are similar
equation of a horizontal line
y = #
equation of a vertical line
x = #
triangle angle sum theorem
<1 + <2 + <3 = 180
exterior angle sum theorem
<1 + <2 + < 4
circumcenter
point of concurrency between the perpendicular bisectors, can be outside, inside, or on the triangle, if the triangle is right it lies on the midpoint of the hypotenuse
incenter
point of concurrency between the angle bisectors, will always be inside the circle
orthocenter
point of concurrency between the altitudes of each triangles
if the triangle is acute, the circumcenter and orthocenter are inside the triangle, if it is obtuse, they are outside the triangle, if it’s right the circumcenter is on the midpoint of the hypotenuse, and the orthocenter is on the vertex of the right angle
in a triangle, the third side must be more than the difference of the two sides but less than the sum of the other two sides
the shortest side is opposite to the smallest angle, the longest side is opposite to the longest angle
when finding angles for a inscribed quadrilateral –> opposite angles
seg 1 /a = a /seg 2, leg/h = h/leg projection
