Question

1. two concentric circles share the same center but have radii of 3.5 and 7. what is the scale factor of the dilation needed to map the smaller circle into the larger circle?
2. trapezoid pqrs has vertices p(6, 6), q(4, 6), r(3, 0), s(1, 0). after a dilation with a scale factor k = 1/2, what are the coordinates of r?
3. a line segment has endpoints p(2, 3) and q(6, 9). after a dilation with k = 1/2, what are the new coordinates of q?
4. a rectangle with vertices p(1, 1), q(3, 1), r(3, 4), and s(1, 4) is dilated to p(2, 2), q(6, 2), r(6, 8), and s(2, 8). what is the dilation factor?
5. a smaller circle with a radius of 2 and center (-2, 4) needs to be transformed into a larger circle with a radius of 10 and center (3, -2). what transformations are needed?
6. a square has vertices at a(-2, -2), b(-2, 2), c(2, 2), and d(2, -2). if it is dilated with a scale factor of 4 and the center of dilation is at (0, 0), what are the coordinates of c?
7. a model of a sports arena is being created using a scale factor of 0.24. if the actual arena is 100 meters long, what is the length of the model?

Explanation:

Step1: Recall the formula for scale factor in dilation

The scale factor \(k\) for a dilation is given by \(k=\frac{\text{image dimension}}{\text{original dimension}}\).

Step2: Calculate the scale factor for each problem

• Problem 7:

The radii of the two circles are \(r_1 = 3.5\) and \(r_2=7\). Using the formula \(k=\frac{r_2}{r_1}\), we have \(k = \frac{7}{3.5}=2\).

• Problem 9:

Given a dilation with scale factor \(k=\frac{1}{2}\) and a point \(R(3,0)\). The coordinates of the dilated point \(R'\) are found by multiplying each coordinate of \(R\) by \(k\). So \(x'=3\times\frac{1}{2}=1.5\) and \(y'=0\times\frac{1}{2} = 0\), so \(R'=(1.5,0)\).

• Problem 10:

Given a dilation with scale factor \(k=\frac{1}{2}\) and a point \(Q(6,3)\). The coordinates of the dilated point \(Q'\) are \(x'=6\times\frac{1}{2}=3\) and \(y'=3\times\frac{1}{2}=1.5\), so \(Q'=(3,1.5)\).

• Problem 11:

Take a point \(P(1,1)\) and its image \(P'(2,2)\). Using the formula \(k=\frac{x_{P'}}{x_P}=\frac{y_{P'}}{y_P}\), we get \(k = 2\).

• Problem 12:

The center of the circle changes from \((-2,4)\) to \((3,-2)\), which requires a translation. The radius changes from \(r_1 = 2\) to \(r_2 = 10\), so \(k=\frac{10}{2}=5\) (a dilation). So we need dilation and translation.

• Problem 13:

Given a square with a vertex \(C(2,2)\) and a scale factor \(k = 4\) and center of dilation \((0,0)\). The coordinates of \(C'\) are \(x'=2\times4 = 8\) and \(y'=2\times4=8\), so \(C'=(8,8)\) (but if we consider the general formula for dilation \((x,y)\to(kx,ky)\) with \(k = 4\) and center \((0,0)\)).

• Problem 14:

Given a scale factor \(k=0.04\) and actual length \(L = 800\) meters. Using the formula \(l=k\times L\), we have \(l=0.04\times800=32\) meters.

Answer:

1. a. 2
2. d. \((1.5,0)\)
3. c. \((3,1.5)\)
4. b. 2
5. d. Dilation and translation
6. b. \((8,8)\)
7. b. 32 meters