Which of the following represents the translation of A (1, −2) along the vector <−5, 1> and then the vector <3, 0>? A. A (1, −2) → A ′(2, −7) → A ″(6, −7)
B. A (1, −2) → A ′(−4, −1) → A ″(−1, −1)
C. A (1, −2) → A ′(−5, 1) → A ″(3, 0)
D. A (1, −2) → A ′(−5, −2) → A ″(−15, 0)

B. A (1, −2) → A ′(−4, −1) → A ″(−1, −1)

Step-by-step explanation:

When you see anything in between <> (let's say <a, b> are the numbers), it basically means <x+a, y+b>.

So if you replace a with -5, b with 1, and x and y with (1, -2) you get...

= (1 + (-5), -2 + 1)

= (1-5, -1)

= (-4, -1) > Those are the coordinates of the new point.

But wait, it says that we now have to translate along the vector <3, 0>

So (-4, -2) become the new (x, y) coordinates and <3, 0> become the new a, b

= (-4+3, -1 + 0)

= (-1, -1)

That means that the coordinates of point A'' are (-1, -1)

Another way you could do this is by combining the two translations to just do one translation. It'll skip over the middle step, but will still give you the correct answer.

You would combine them by adding the translations of the x-coordinates together and the y-coordinates together

= <-5 + 3, 1 + 0 >

= <-2, 1>

Now you translate the point by this vector and...

= (1 - 2, -2 + 1)

= (-1, -1)

Tada! It works!

Therefore, the answer is B.

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.

answer: a

step-by-step explanation: the answer is a due to the rule of symmetry; that a line of symmetry must divide a figure into 2 pieces that are the exact same, only reflected over a line of symmetry. b and c do not have lines of symmetry and would not divide the figure evenly, thus making the answer a.