Question

1. graph any triangle and translate it in any direction. draw translation vectors for each vertex of the triangle. is there a geometric relationship between all the translation vectors? explain why this makes sense in terms of the slope of the line.
2. point ( p(4, -2) ) undergoes a translation given by ( (x, y) \to (x + 3, x - a) ), followed by another translation ( (x, y) \to (x - b, x + 7) ) to produce the image of ( p(-5, 8) ). find the values of ( a ) and ( b ) and point ( p ).

Explanation:

Step1: Apply first translation to P

First translation: $(4, -2) \to (4+3, 4 - a) = (7, 4 - a) = P'$

Step2: Apply second translation to P'

Second translation: $(7, 4 - a) \to (7 - b, 7 + 7) = (7 - b, 14)$

Step3: Set equal to P'' for y-coordinate

$14 = 8$? No—correction: Second translation y is x+7 (x of P' is 7), so y=7+7=14? Wait, P'' y is 8: $4 - a + 7 = 8$ → $11 - a = 8$ → $a=3$

Step4: Find P' with a=3

$P'=(7, 4 - 3)=(7,1)$

Step5: Set x-coordinate equal to P''

$7 - b = -5$ → $b=12$

Answer:

a=3, b=12, P'(7,1)