midpoint, classify triangle, partition line segment, applications, and dilations
find the coordinates of the midpoint of the segment whose endpoints are (-2, 3) and (4, -3)
M = (1, 0)
what is the midpoint formula
M = (x1 + x2 / 2, y1 + y2 / 2)
write the equation of the line that is the perpendicular bisector of the line segment joining the points (-2, 4) and (4, -6)
M AB = (1, -1) mAB = -5/3 y + 1 = 3/5 (x-1)
the endpoints of DEF are D (1,4) and F (16, 14). Determine and state the coordinates of point E, if DE:EF = 2:3
XE=7 YE=8 E = (7,8)
What happens if we dilate an entire line, but the center of dilation is NOT on the line?
a dilation takes a line NOT passing through the center of the dilation to a parallel line. because corresponding lines are parallel and angle measures are preserved in dilations
a dilation of -1/3 with center of dilation as the origin
a scale factor that is negative reflects that image over the center of dilation
scalene
all sides have different distances
In circle O, a diameter has endpoints (-5, 4) and (3, -6). what is the length of the diameter? What are the coordinates of the center of the circle
d = 2√41 units M (center) = (-1, -1)
median
find the midpoint of AB, use the coordinates with the coordinates of C to make the slope, use the midpoint of AB and the new slope to make the equation of the line
what do you do when you get points on a line that have already been dilated by 1/2 and you have to find the original distance
find the original points by multiplying by 2 find the distance with those points
what is midpoint formula used for
finding midpoints finding endpoints prove lines bisect each other find the center of a circle
rules for a dilation of 1/2 with the center of dilation as the origin
if 0<k<1 you get an image smaller than the pre-image
obtuse
if a2 + b2 < c2
right
if a2 + b2 = c2
acute
if a2 + b2 > c2
altitude
inverse reciprocal of slope of AB with point c to make equation of line
scale factor
k
when describing a transformation of a dilation what do you say
k = ? - x1(k) = x2 center of dilation is (?,?) - draw a parallel line from each point from each shape
how do you explain why two lines are parallel
lines are parallel if and because they have the same slope
when finding the image parallel of an equation of the line that is centered at the origin what do you have to do (multi choice)
make sure it is not on the line by putting in 0,0 (to see if the lines are parallel) find m and use the same one in the new equation try the equations they give you and see which one gets the same m but a different y - intercept
when solving with zero
make sure you put in zero so the numbers that are multiplied, divided, subtracted, or added with zero are correct
how do you multiply fractions
multiply the numerators and then multiply the denominators or multiply the whole number by the fraction and then on the calculator hit the second button then the F <-> d button
what do dilations create when dealing with figures
parallel segments
rules of a dilation of 2 with center of dilation as the origin
scale factor k > 1, produces larger image than pre-image center of dilation, the pre-image and image are always collinear
what does midpoint formula represent
the average of the points
which side is c2
the biggest ex. √29² + √72² = √137²
what happens if we dilate an entire line
the dilation of the line, with the center of dilation on the line, leaves the line unchanged (we get the same line again). The scale factor is of no importance
equilateral
the distance of all three sides are equal
k = -1/3
the negative sign reflects and the fraction makes the image shorter
what are the steps of finding the equation of a perpendicular bisector, if you get the coordinates of both endpoints?
think of properties the perpendicular bisector must have 1. must pass through the midpoint of AB 2. find the slope of the line and take the inverse reciprocal 3. find the midpoint of the endpoints 4. write the point slope form of the equation of the line with the midpoints and the inverse reciprocal of the slope
when solving radicals you
use the number that is the highest radical as the first radical and the answer is a whole number then a radical
rule of rise and run technique
we don't simplify this slope in anyway because it is actually describing the VECTOR <5,2>
when do you use the rise and run technique
when there is a dilation of k and the center is off the origin
write the equation of AB
write the point slope form of a line
how do you solve for a point when given the other point and the midpoint algebraically
write the points in the order in the picture and find the delta x and y and draw arrows where the points are. Then add delta x and y to the midpoint to get the next point
A (2, -1) B (x, 5) AB (3, square root 5)find the x coordinate for point B
x = 5 or x = -1
perpendicular bisector of an undefined slope
y = c and the equation is going through the midpoint
what do you do when you get the equation of a line and another line you have to *dilate* is mapped onto that line
you find the y=mx+b form of the line and get the x and y coordinates. You then keep m the same and multiply x and y by the k if it is centered at the origin.
dividing a line segment into a given ratio
you make a fraction out of the ratio you are given ex. 2:3 = 2/5 then multiply the fraction by delta x then add the start point of x repeat the same with y the two points you get are the point _
when dilation is on the line
you put in 0,0 and the solution is equal nothing changes
when the dilation is off the line
you put in 0,0 and the solutions do not equal each other the image and pre-image are parallel
rise and run technique
1. find the slope from the center of dilation to the point 2. multiply the rise and run by k 3. add that value to the original point and mark it on the coordinate plane
isosceles
2 sides have the same distance
Midpoint M of AB has coordinate (6,5). If the coordinates of A are (4,1) what are the coordinates of B?
A = (4, 1) Delta x = 2, Delta Y = 4 Mab = (6,5) 6+2 = 8, 5+4 = 9 B = (8, 9)
