Question

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o center (5,6) and scale factor 2 o center (3,8) and scale factor 2
o center (5,6) and scale factor 3 o center (3,8) and scale factor 3
o center (5,6) and scale factor ⅓ o center (3,8) and scale factor ⅓

Explanation:

Step1: Recall scale - factor formula

For a dilation with center \((h,k)\) and scale - factor \(k\), the transformation of a point \((x,y)\) to \((x',y')\) is given by \(x'=h + k(x - h)\) and \(y'=k + k(y - k)\). We can also use the property that if we know the relationship between corresponding points of the original and dilated figures, we can find the center and scale - factor.
Let's assume two corresponding points \(A(x_1,y_1)\) on the original figure and \(A'(x_2,y_2)\) on the dilated figure. The formula for the center \((h,k)\) and scale - factor \(k\) in terms of two corresponding points is \(k=\frac{x_2 - h}{x_1 - h}=\frac{y_2 - k}{y_1 - k}\).
However, an alternative way is to look at the distance between corresponding points and a potential center point.
Let's assume the center of dilation is \((h,k)\). If we take a point \((x_1,y_1)\) on the original figure and its corresponding point \((x_2,y_2)\) on the dilated figure, the vector from the center \((h,k)\) to \((x_1,y_1)\) is \(\vec{v_1}=(x_1 - h,y_1 - k)\) and the vector from the center \((h,k)\) to \((x_2,y_2)\) is \(\vec{v_2}=(x_2 - h,y_2 - k)\), and \(\vec{v_2}=k\vec{v_1}\).
Let's assume some easy - to - identify points on the two triangles. Suppose a point on the small triangle is \((0,0)\) (assuming one of the vertices is at the origin for simplicity of calculation, we can choose any pair of corresponding points). Let's assume the corresponding point on the large triangle is \((15,18)\).
If the center of dilation is \((h,k)\) and scale - factor is \(k\), then \(15=h + k(0 - h)=h(1 - k)\) and \(18=k + k(0 - k)=k(1 - k)\).
Another way is to note that if we consider the ratio of the side - lengths of the two similar triangles. Measure the lengths of corresponding sides. Let's assume we measure the length of a side of the small triangle \(l_1\) and the length of the corresponding side of the large triangle \(l_2\). The scale - factor \(k=\frac{l_2}{l_1}\).
By measuring (using the grid), if we assume one side of the small triangle has a length of 5 units and the corresponding side of the large triangle has a length of 15 units, the scale - factor \(k = 3\).
To find the center of dilation, we can use the property that the lines connecting corresponding points intersect at the center of dilation. By drawing the lines connecting corresponding vertices of the two triangles, we find that they intersect at the point \((5,6)\).

Answer:

Center \((5,6)\) and scale factor \(3\)