Question

Is them being the same shape mean they are congruent? Need explanation

Answer 1

In order to determine if the given figures are congruent, we can shown that the length of three pair of segments are equal.

For instance, you can notice that the length of AB=2 and the length of DE=2.

To calculate the length of the other segments, use the following formula:

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

For intance, for segment BC, you have (x1,y1) = (2,0) and (x2,y2) = (3,4). Then, length of the segment is:

`d=\sqrt[]{(3-2)^2+(4-0)^2}=\sqrt[]{1^2+4^2}=\sqrt[]{1+16}=\sqrt[]{17}`

Now, for segment EF, (x1,y1) = (-1,2) and (x2,y2) = (3,1). The length of the segment is:

`\begin{gathered} d=\sqrt[]{(3-(-1))^2+(2-1)^2}=\sqrt[]{(3+1)^2+(1)^2} \\ d=\sqrt[]{(4)^2+(1)^2}=\sqrt[]{16+1}=\sqrt[]{17} \end{gathered}`

As you can notice the length of BC is equal to the length of EF.

For the segment CA, you have (x1,y1) = (0,0) and (x2,y2) = (3,4). The length of the segments is;

`\begin{gathered} d=\sqrt[]{(3-0)^2+(4-0)^2}=\sqrt[]{(3)^2+(4)^2}^{} \\ d=\sqrt[]{9+16}=\sqrt[]{25}=5 \end{gathered}`

Now, for segment FD, (x1,y1) = (-1,2) and (x2,y2) = (3,-1). Then, the length of the segment is;

`\begin{gathered} d=\sqrt[]{(3-(-1))^2+(-1-2)^2}=\sqrt[]{(3+1)^2+(-3)^2} \\ d=\sqrt[]{(4)^2+(-3)^2}=\sqrt[]{16+9}=\sqrt[]{25}=5 \end{gathered}`

As you can notice, the length of segment CA is equal to the length of segment FD.

Hence, you can conclude that AB=DE, BC = EF and CA=FD. It means that the given figures are congruent by the side-side-side theorem (SSS).