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.

Explanation:

Step1: Choose original triangle vertices

Let original triangle vertices be $A(1,1)$, $B(3,1)$, $C(2,3)$.

Step2: Define translation vector

Use translation vector $\vec{v}=(2,1)$ (right 2, up 1).

Step3: Find translated vertices

$A'(1+2,1+1)=(3,2)$, $B'(3+2,1+1)=(5,2)$, $C'(2+2,3+1)=(4,4)$.

Step4: Calculate translation vectors

$\vec{AA'}=(3-1,2-1)=(2,1)$, $\vec{BB'}=(5-3,2-1)=(2,1)$, $\vec{CC'}=(4-2,4-3)=(2,1)$.

Step5: Analyze slope relationship

Slope of each vector: $\frac{1}{2}$ (same for all).

Answer:

All translation vectors are equal and parallel (same direction and magnitude). Their slopes are identical because translation applies the same horizontal and vertical change to every vertex, so the rise-over-run ratio (slope) for each vector is the same.