Question

Is quadrilateral jklm the result of a dilation of quadrilateral abcd by a scale factor of 2? why or why not?

Answer 1

No. Because sides JM and KL have different slopes from sides AD and BC .

The formula of a slope:

$m=\dfrac{y_2-y_1}{x_2-x_1}$

For AD:

We have the points A(2, -2) and D(1, -4). Substitute:

$m=\dfrac{-4-(-2)}{1-2}=\dfrac{-2}{-1}=2$

For JM:

We have the points J(4, -4) and M(2, -9). Substitute:

$m=\dfrac{-9-(-4)}{2-4}=\dfrac{-5}{-2}=2.5$

$2\neq2.5$

----------------------------------------

Another argument.

No. Because the MJ is not twice as long as AD.

The formula of the length of a segment:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The length of AD:

$|AD|=\sqrt{(1-2)^2+(-4-(-2))^2}=\sqrt{(-1)^2+(-2)^2}=\sqrt{1+4}=\sqrt5$

The length of MJ:

$|MJ|=\sqrt{(2-4)^2+(-9-(-4))^2}=\sqrt{(-2)^2+(-5)^2}=\sqrt{4+25}=\sqrt{29}$

$\sqrt{29}\neq2\sqrt5$