Question

Choose the explanation that shows why ΔADE is similar to ΔABC.

A. A dilation with center (0, 0) and scale factor 1/2 maps ΔABC to ΔADE.
B. A dilation with center (0, 0) and scale factor 2 maps ΔABC to ΔADE.
C. A translation 4 units right and 4 units up maps ΔABC to ΔADE.
D. A rotation 90° about the origin and a translation 2 units up maps ΔABC to ΔADE.

Answer 1

Hello,

First of all,
C. and D. are both incorrect because they apply translations to ABC; translations will misalign ADE with ABC.

Next,
A. is incorrect because a dilation of 1/2 will place D at (0,3) from its original point of (0,2). Also, the dilation will place E at (6,0) from its original point of (4,0). The triangle ADE would be too small to map ABC.

Finally,

B. Is the correct answer because a dilation of 2 will place D at (0,4)—where B is—and will place E at (8,0), where C is.
Choice B. maps ADE onto ABC

Sorry for that much information, but I think it’s useful. XD

Answer 2

The correct explanation is A. A dilation with center (0, 0) and scale factor 1/2 maps ΔABC to ΔADE.

A dilation is a transformation that enlarges or reduces a shape by a certain scale factor, while keeping the shape the same. In this case, the scale factor is 1/2, which means that ΔADE is half the size of ΔABC.

The other options are incorrect:

B. A dilation with center (0, 0) and scale factor 2 maps ΔABC to ΔADE is incorrect because ΔADE would be twice the size of ΔABC, not half the size.

C. A translation 4 units right and 4 units up maps ΔABC to ΔADE is incorrect because a translation moves the shape without changing its size or orientation.

D. A rotation 90° about the origin and a translation 2 units up maps ΔABC to ΔADE is incorrect because a rotation changes the orientation of the shape, not its size.

Therefore, the only explanation that shows why ΔADE is similar to ΔABC is that a dilation with center (0, 0) and scale factor 1/2 maps ΔABC to ΔADE.

To see this visually, we can look at the image you provided. The image shows two parallel lines, with ΔABC above the lines and ΔADE below the lines. The scale factor of 1/2 is evident from the fact that ΔADE is half the height of ΔABC.

Additionally, we can see that the two triangles have the same angles. This is because a dilation preserves the angles of the original shape. Therefore, ΔADE is similar to ΔABC.