Question

(a) draw each of the images described in the steps below. step 1: first, starting with the given figure, perform a dilation centered at the origin with a scale factor of \\(\frac{1}{4}\\). step 2: then from the resulting figure, perform a dilation centered at the origin with a scale factor of 3. (b) fill in the blank to describe the situation. performing steps 1 and 2 is equivalent to a single dilation centered at the origin with a scale factor of \\(\square\\).

Explanation:

Part (a) (Step-by-Step Format)

Step1: Identify original triangle vertices

Original vertices (from graph): $(3,12)$, $(14,12)$, $(14,16)$

Step2: Apply 1st dilation (scale $\frac{1}{4}$)

Multiply each coordinate by $\frac{1}{4}$:
$(3\times\frac{1}{4}, 12\times\frac{1}{4}) = (\frac{3}{4}, 3)$
$(14\times\frac{1}{4}, 12\times\frac{1}{4}) = (\frac{7}{2}, 3)$
$(14\times\frac{1}{4}, 16\times\frac{1}{4}) = (\frac{7}{2}, 4)$

Step3: Apply 2nd dilation (scale 3)

Multiply each new coordinate by 3:
$(\frac{3}{4}\times3, 3\times3) = (\frac{9}{4}, 9)$
$(\frac{7}{2}\times3, 3\times3) = (\frac{21}{2}, 9)$
$(\frac{7}{2}\times3, 4\times3) = (\frac{21}{2}, 12)$
Plot these final points to draw the image.

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Part (b) (Step-by-Step Format)

Step1: Combine scale factors

Multiply the two scale factors.
$\frac{1}{4} \times 3$

Step2: Calculate equivalent scale factor

Simplify the product.
$\frac{3}{4}$

Answer:

$\frac{3}{4}$