∆PQR has vertices at P(2, 4), Q(3, 8) and R(5, 4). A dilation and series of translations map ∆PQR to ∆ABC, whose vertices are A(2, 4), B(5.5, 18), and C(12.5, 4). What is the scale factor of the dilation in the similarity transformation?

Question

∆PQR has vertices at P(2, 4), Q(3, 8) and R(5, 4). A dilation and series of translations map ∆PQR to ∆ABC, whose vertices are A(2, 4), B(5.5, 18), and C(12.5, 4). What is the scale factor of the dilation in the similarity transformation?

Answer 1

To find the scale factor of the dilation in the similarity transformation, we can compare the corresponding side lengths of the two triangles.

The corresponding side lengths are:
PQ: distance between P(2, 4) and Q(3, 8)
AB: distance between A(2, 4) and B(5.5, 18)

Using the distance formula, we have:
PQ = sqrt((3 - 2)^2 + (8 - 4)^2) = sqrt(1 + 16) = sqrt(17)
AB = sqrt((5.5 - 2)^2 + (18 - 4)^2) = sqrt(14.5^2 + 14^2) = sqrt(210.25 + 196) = sqrt(406.25) = 20.156

Therefore, the scale factor of the dilation is:
AB / PQ = 20.156 / sqrt(17) ≈ 3.879

Answer 2

To find the scale factor of the dilation in the similarity transformation, we need to compare the corresponding side lengths of the two triangles.

The corresponding side lengths of ∆PQR and ∆ABC are:
- PQ and AB
- QR and BC
- RP and CA

Let's calculate the lengths of these sides using the distance formula:

For ∆PQR:
PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - 2)^2 + (8 - 4)^2) = sqrt(1 + 16) = sqrt(17)
QR = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((5 - 3)^2 + (4 - 8)^2) = sqrt(4 + 16) = sqrt(20)
RP = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((2 - 5)^2 + (4 - 4)^2) = sqrt(9) = 3

For ∆ABC:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((5.5 - 2)^2 + (18 - 4)^2) = sqrt(3.5^2 + 14^2) = sqrt(12.25 + 196) = sqrt(208.25)
BC = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((12.5 - 5.5)^2 + (4 - 18)^2) = sqrt(7^2 + (-14)^2) = sqrt(49 + 196) = sqrt(245)
CA = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((2 - 12.5)^2 + (4 - 4)^2) = sqrt((-10.5)^2 + 0) = sqrt(110.25) = 10.5

Now, let's compare the corresponding side lengths:
(scale factor) = (length of corresponding side in ∆ABC) / (length of corresponding side in ∆PQR)

For PQ/AB:
scale factor = AB / PQ = sqrt(208.25) / sqrt(17)

For QR/BC:
scale factor = BC / QR = sqrt(245) / sqrt(20)

For RP/CA:
scale factor = CA / RP = 10.5 / 3

So, the scale factor of the dilation in the similarity transformation is the average of the individual scale factors for each pair of corresponding side lengths:

(scale factor) = (AB / PQ + BC / QR + CA / RP) / 3

Plug in the values and calculate the scale factor.

Answer 3

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