Question

In the following graph we observe line segment AB being mapped by translation to line segment A'B'. A. In two or more sentences, explain what test you could perform to check the validity of this translation. B. Prove mathematically that the translation is valid using the test you mentioned above.

Answer 1

You could use the distance formula to prove its the same line.
square root of (x2-x1)^2+(y2-y1)^2
(5-3)^2+(6-9)^2
4+9
square root of 13


(-2- -4)^2+(3-6)^2
4+9
square root of 13

this proves it is the same line.

Answer 2

Answer:

We are given that the line AB with coordinates (-4,6) and (-2,3) is translated to the line A'B' with co-ordinates (3,9) and (5,6).

1. In order to check whether the translation is valid,

We will use the distance formula in order to find the distance between the two points of the lines.

If they are same, then we get that the translation applied will be valid.

2. We know that the distance formula for two points $(x_{1},y_{1})$ and  $(x_{2},y_{2})$ is given by $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$.

So, the distance of AB is,

$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$ = $\sqrt{(-2-(-4))^{2}+(3-6)^{2}}$ = $\sqrt{2^{2}+(-3)^{2}}$ = $\sqrt{4+9}$ = $\sqrt{13}$

Also, the distance of A'B' is,

$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$ = $\sqrt{5-3)^{2}+(6-9)^{2}}$ = $\sqrt{2^{2}+(-3)^{2}}$ = $\sqrt{4+9}$ = $\sqrt{13}$

Thus, we see that the distance between the two points of each line AB and A'B' is $\sqrt{13}$ i.e. same.

Hence, the given translation is valid.