Graph a triangle abc and perform a translation of (x + 4, y − 3) to create triangle a′b′c′.

describe the transformation using words. make sure you refer to the characteristics and the coordinates.
draw a line through points a and a′ and through points b and b′. what do you notice about the lines you drew? do you think you would notice the same characteristics if you drew another line through points c and c′? how do you know?

answer:

step-by-step explanation:

let's study an example problem

find the image a'a

′

a, prime of a(4,-7)a(4,−7)a, left parenthesis, 4, comma, minus, 7, right parenthesis under the transformation t_{(-10,5)}t

(−10,5)

​

t, start subscript, left parenthesis, minus, 10, comma, 5, right parenthesis, end subscript.

solution

the translation t_{(\teald{-10},\maroond{5})}t

(−10,5)

​

t, start subscript, left parenthesis, start color #01a995, minus, 10, end color #01a995, comma, start color #ca337c, 5, end color #ca337c, right parenthesis, end subscript moves all points \teald{-10}−10start color #01a995, minus, 10, end color #01a995 in the xxx-direction and \maroond{+5}+5start color #ca337c, plus, 5, end color #ca337c in the yyy-direction. in other words, it moves everything 10 units to the left and 5 units up.

now we can simply go 10 units to the left and 5 units up from a(4,-7)a(4,−7)a, left parenthesis, 4, comma, minus, 7, right parenthesis.

we can also find a'a

′

a, prime algebraically:

a'=(4\teald{-10},-7\maroond{+5})=(-6,-2)a

′

=(4−10,−7+5)=(−6,−2)a, prime, equals, left parenthesis, 4, start color #01a995, minus, 10, end color #01a995, comma, minus, 7, start color #ca337c, plus, 5, end color #ca337c, right parenthesis, equals, left parenthesis, minus, 6, comma, minus, 2, right parenthesis

your turn!

problem 1

draw the image of b(6,2)b(6,2)b, left parenthesis, 6, comma, 2, right parenthesis under transformation t_{(-4,-8)}t

(−4,−8)

​

t, start subscript, left parenthesis, minus, 4, comma, minus, 8, right parenthesis, end subscript.

[i need a hint! ]

t_{(\teald{-4},\maroond{-8})}

t, start subscript, left parenthesis, start color #01a995, minus, 4, end color #01a995, comma, start color #ca337c, minus, 8, end color #ca337c, right parenthesis, end subscript\teald{-4}start color #01a995, minus, 4, end color #01a995xx\maroond{-8}start color #ca337c, minus, 8, end color #ca337cyy

problem 2

what is the image of (23,-15)(23,−15)left parenthesis, 23, comma, minus, 15, right parenthesis under the translation t_{(12,32)}t

(12,32)

​

t, start subscript, left parenthesis, 12, comma, 32, right parenthesis, end subscript?

((left parenthesis

,,comma

))right parenthesis

[i need a hint! ]

t_{(\teald{12},\maroond{32})}

t, start subscript, left parenthesis, start color #01a995, 12, end color #01a995, comma, start color #ca337c, 32, end color #ca337c, right parenthesis, end subscript\teald{+12}start color #01a995, plus, 12, end color #01a995xx\maroond{+32}start color #ca337c, plus, 32, end color #ca337cyy

part 2: translating line segments

let's study an example problem

consider line segment \overline{cd}

cd

start overline, c, d, end overline drawn below. let's draw its image under the translation t_{(9,-5)}t

(9,−5)

​

t, start subscript, left parenthesis, 9, comma, minus, 5, right parenthesis, end subscript.

solution

when we translate a line segment, we are actually translating all the individual points that make up that segment.

luckily, we don't have to translate all the points, which are infinite! instead, we can consider the endpoints of the segment.

since all points move in exactly the same direction, the image of \overline{cd}

cd

start overline, c, d, end overline will simply be the line segment whose endpoints are c'c

′

c, prime and d'd

′

d, prime.

part 3: translating polygons

let's study an example problem

consider quadrilateral efghefghe, f, g, h drawn below. let's draw its image, e'f'g'h'e

′

f

′

g

′

h

′

e, prime, f, prime, g, prime, h, prime, under the translation t_{(-6,-10)}t

(−6,−10)

​

t, start subscript, left parenthesis, minus, 6, comma, minus, 10, right parenthesis, end subscript.

solution

when we translate a polygon, we are actually translating all the individual line segments that make up that polygon!

basically, what we did here is to find the images of eee, fff, ggg, and hhh and connect those image vertices.

[i want to see sal solving a similar problem.]

your turn!

problem 1

draw the image of \triangle ijk△ijktriangle, i, j, k under the translation t_{(-5,2)}t

(−5,2)

​

t, start subscript, left parenthesis, minus, 5, comma, 2, right parenthesis, end subscript.

draw \triangle pqr△pqrtriangle, p, q, r.