Question

Figure ABCD [is/is not] congruent to figure JKLM because motions [can/cannot] be used to map figure ABCD onto Figure JKLM . Figure ABCD[is/is not] similar to Figure GHIJK because rigid motions and/or dilation [can/cannot] be used to map Figure ABCD onto Figure JKLM.

Answer 1

Figures are said to be congruent when they have equal size and the same shape.

Figures are said to be similar when they have the same shapes but not the same size.

Here, we are given two figures that have similar shapes but not equal sizes.

To map figure ABCD to figure JKLM, a dilation and a rotation(rigid motion) must be done.

Rigid motions alone can be used to map congruent figures.

Rigid motions alone cannot be used to map map similar figures

Therefore, we have the following:

Figure ABCD is not congruent to figure JKLM because motions cannot be used to map figure ABCD onto figure JKLM

Figure ABCD is similar to figure JKLM because rigid motions and dilation can be used to map figure ABCD onto figure JKLM

Let's find the scale factor of the dilation.

Given the vertices of ABCD:

A(3, 1), B(4, 1), C(2, 4), D(1, 3)

Vertices of JKLM:

J(3, -9), K(3, -12), L(12, -6), M(9, -3)

To find the scale factor, divide the distances of figure JKLM by the correpsonding distance of figure ABCD.

Apply the distance formula:

`d=\sqrt[]{(x2-x1)^2+(y2-y1)^2}`

Thus we have:

`\begin{gathered} AB=\sqrt[]{(4-3)^2+(1-1)^2} \\ AB=\sqrt[]{1^2+0^2} \\ AB=1 \end{gathered}`
`\begin{gathered} JK=\sqrt[]{(3-3)^2+(-12-(-9))^2} \\ JK=\sqrt[]{0^2+(-12+9)^2} \\ JK=\sqrt[]{0^2+(-3)^2} \\ JK=\sqrt[]{3^2} \\ JK=3 \end{gathered}`

Thus, the scale factor is:

`\text{Scale factor=}(JK)/(AB)=(3)/(1)=3`

Therefore, the scale factor that from figure ABCD to figure JKLM is 3

ANSWER:

• Figure ABCD, ,is not, , congruent to figure JKLM because motions ,cannot, be used to map figure ABCD onto figure JKLM

,

• Figure ABCD ,is, , similar to figure JKLM because rigid motions and dilation ,can , be used to map figure ABCD onto figure JKLM

,

• The scale factor that from figure ABCD to figure JKLM is ,3