Question

∆ABC and ∆PQR are similar. ∆ABC is dilated by a scale factor of 1.25 and rotated 45° counterclockwise about point B to form ∆PQR. The side lengths of ∆ABC are , 5 units; , 4.2 units; and , 4 units. Match each side of ∆PQR to its length.

Answer 1

After dilating
`\( \Delta ABC \)` by a scale factor of 1.25 and rotating it 45° counterclockwise about point B, the side lengths of
`\( \Delta PQR \)` are PQ = 6.25 units, QR = 5.25 units, and PR = 5 units.

Let's denote the side lengths of
`\( \Delta ABC \)` as
`\( AB, BC, \)` and
`\( AC \)`, and the corresponding side lengths of
`\( \Delta PQR \)` as PQ, QR and PR.

Given that
`\( \Delta ABC \)` is dilated by a scale factor of 1.25 and rotated 45° counterclockwise about point B, we can determine the side lengths of
`\( \Delta PQR \)` as follows:

1. Dilating by a scale factor of 1.25:

-
`\( PQ = 1.25 * AB \)`

-
`\( QR = 1.25 * BC \)`

-
`\( PR = 1.25 * AC \)`

2. Rotating 45° counterclockwise about point B:

- PQ, QR, and PR are now the rotated side lengths.

Now, let's substitute the given side lengths of
`\( \Delta ABC \)` to find the corresponding side lengths of
`\( \Delta PQR \)`:

Given:

- AB = 5 units

- BC = 4.2 units

- AC = 4 units

Substitute these values:

1. Dilating by a scale factor of 1.25:

-
`\( PQ = 1.25 * 5 = 6.25 \)` units

-
`\( QR = 1.25 * 4.2 = 5.25 \)` units

-
`\( PR = 1.25 * 4 = 5 \)` units

2. Rotating 45° counterclockwise about point B:

- The side lengths PQ, QR, and PR after rotation are the final side lengths.

So, the matched side lengths of
`\( \Delta PQR \)` are:

- PQ = 6.25 units

- QR = 5.25 units

- PR = 5 units

Answer 2

This is the concept of transformation of figures, given that ΔABC is similar to ΔPQR, the sides of ABC are 5 units, 4.2 units and 4 units. Since the two triangles are similar and PQR is the image of ABC under the dilation 1.25, the sides of PQR will be:
(5*1.25),(4.2*1.25),(4*1.25)
this will give us:
6.25, 5.25, 5
The length of the sides of PQR are 6.25 units, 5.25 units and 5 units