What are certainty, proof of intuition and proof of contrapositive?

self check problem 5. rotate the figure with vertices a(3, 3), b(2, 6) and c(-1, 1) 90º counter-clockwise about the origin. solution

essential question: how can the coordinate plane help me understand properties of reflections, translations and rotations?

a translation is a transformation where all points of the figure are moved the same distance in the same direction. the distance and direction are indicated by a ray sometimes called the translation vector. a vector is a quantity that has both length and direction, and can be thought of as a line segment with a starting point and an endpoint. a translation is an isometry, so the image of a translated figure is congruent to the preimage.

often, the rule for the translation will be given as (x,  y) → (x  +  a,  y  +  b). this is a shift  a  units in the  x-direction (horizontally) and  b  units in the  y-direction (vertically). for instance, in the coordinate plane below the translation (x,  y) → (x  + 4,  y  – 2) shifts each point 4 units to the right and 2 units down.

such translations are isometries and shift the preimage without changing its size. however, if the rule contains a constant multiplied by  x  or  y, the transformation is no longer a translation. the multiplier changes the size of the image making it a non-rigid transformation. for example, consider the line segment with endpoints (-2, 1) and (3, 4). to translate the line segment by the rule (x  + 2,  y  – 1), add 2 to both  x  coordinates and subtract 1 from both  y  coordinates. 

(-2, 1) → (-2 + 2, 1 – 1) = (0, 0)
(3, 4) → (3 + 2, 4 – 1) = (5, 3)

the preimage is shown in blue and the image is shown in red. the preimage is shifted 2 units to the right and 1 unit down, without changing its size or shape.

if the same preimage with endpoints (-2, 1) and (3, 4) is transformed according to the rule (x,  y) → (2x  – 3, 2y), the image will not be the same size as its preimage. multiply each  x-coordinate of the preimage by 2 and subtract 3, and multiply each  y-coordinate of the preimage by 2.

(-2, 1) → (2(-2) – 3, 2(1)) = (-7, 2)
(3, 4) → (2(3) – 3, 2(4)) = (3, 8)

the size of the preimage is not preserved.

when two or more transformations are combined to produce a single transformation, the result is called a composition of the transformations. the transformation of two or more isometries is also an isometry.

the rules for different transformations are summarized here. to perform a composition of transformations, apply the rules one at a time.

for instance, if the point (2, 3) is reflected in the  x-axis and then translated by the rule (x,  y) → (x  – 1,  y  – 3), first apply the rule for a reflection in the  x-axis, and then apply the translation rule.

reflection in the  x-axis: (x,  y) → (x, -y), (2, 3) → (2, -3)
translation: (x,  y) → (x  – 1,  y  – 4), (2, -3) → (2 – 1, -3 – 3) = (1, -6)

example 1.  describe the translation that maps the blue figure onto the red figure.

solution: the drawing below shows the translation vectors for two pairs of corresponding vertices. each point on the preimage is moved 3 units to the left and 4 units down.  x,  y) → (x  – 3,  y  – 4)

example 2.  use the transformation rule (x,  y) → (x  + 5,  y  + 2) to answer the following question.

part a: determine the image of a(2, 3). solution: add 5 to the  x-coordinate and add 2 to the  y-coordinate: a(2, 3) → a′(2 + 5, 3 + 2) = (7, 5)
part b: determine the image of q(1, 6). solution: add 5 to the  x-coordinate and add 2 to the  y-coordinate: q(1, 6) → q′(1 + 5, 6 + 2) = (6, 8)
part c: determine the preimage d for d′ (1, 7). solution: the image of d is already known, set up an equation for  x  and an equation for  y  and solve:
x  + 5 = 1
x  = -4

y  + 2 = 7
y  = 5

the preimage of (1, 7) is (-4, 5)

part d: is the transformation an isometry? solution: yes, there is no constant multiplied by the  x  coordinate or the  y  coordinate so the transformation is an isometry.

example 3.  does abcd → fehg (x,  y) → (x  + 4,  y)? if yes, explain how you know. if not, identify the correct transformation.

a. no. in order for the rule to be correct, it would have to map all points on the preimage by the same translation vector. the rule (x,  y) → (x  + 4,  y) maps point b and c only, it does not apply to the entire figure. abcd → fehg by a reflection in the  y-axis.

example 4.  the triangle with vertices a(-1, -3), b(-4, -1), and c(-6, -4) is translated by the rule (x,  y) → (x  + 10,  y) and then reflected over the  x-axis. determine the image ∆a′′b′′c′′.

solution: first apply the translation rule by adding 10 to each  x-coordinate and making no changes to each  y-coordinate.