Question

circle a and circle b are shown below. complete the paragraph to prove circle a is similar to circle b. to map circle a onto circle b, first apply a dilation with center a and a scale factor of to circle a. then, translate the dilated circle 8 units to the and 9 units

Explanation:

Step1: Find radius of Circle A

From the grid, Circle A has a radius of 2 units (distance from center to edge).

Step2: Find radius of Circle B

Circle B has a radius of 10 units (e.g., from center (1, -2) to (9, -2) is 8? Wait, no, let's check grid. Wait, Circle A: center at (-7,7)? Wait, no, looking at the grid, Circle A: center is at (-7,7)? Wait, no, the red circle: from x=-9 to x=-5, so diameter 4, radius 2. Circle B: from x=-7 to x=9, that's 16 units? Wait, no, maybe better: distance from center of B (1, -2) to (9, -2) is 8, so radius 8? Wait, no, the blue circle: top at y=6, center at (1, -2), so vertical distance 8, so radius 8. Wait, Circle A: center at (-7,7), radius: from y=5 to y=9? Wait, no, red circle: center at (-7,7), and it goes from x=-9 to x=-5 (so width 4, radius 2), y from 5 to 9 (height 4, radius 2). So radius of A is 2, radius of B: let's see, center of B is (1, -2). The blue circle goes to x=9 (so 9 - 1 = 8), y=6 (6 - (-2) = 8), y=-10 (-2 - (-10) = 8), x=-7 (1 - (-7) = 8). So radius of B is 8. So scale factor: 8 / 2 = 4. Wait, no, 8/2=4? Wait, 24=8. So scale factor is 4. Then translation: from center of A (-7,7) to center of B (1, -2). The horizontal change: 1 - (-7) = 8 (right), vertical change: -2 - 7 = -9 (down). So first dilation with scale factor 4 (since 24=8, radius of B). Then translate 8 units right and 9 units down.

Step1: Determine radius of Circle A

From the grid, Circle A's diameter spans 4 units (e.g., from \( x = -9 \) to \( x = -5 \)), so radius \( r_A = \frac{4}{2} = 2 \) units.

Step2: Determine radius of Circle B

Circle B's diameter spans 16 units? Wait, no, center of B is \( (1, -2) \). Distance from center to \( x = 9 \) is \( 9 - 1 = 8 \), so radius \( r_B = 8 \) units.

Step3: Calculate scale factor

Scale factor \( k = \frac{r_B}{r_A} = \frac{8}{2} = 4 \).

Step4: Determine translation

Center of A: \( (-7, 7) \), center of B: \( (1, -2) \). Horizontal change: \( 1 - (-7) = 8 \) (right). Vertical change: \( -2 - 7 = -9 \) (down).

Answer:

To map circle A onto circle B, first apply a dilation with center A and a scale factor of \(\boldsymbol{4}\) to circle A. Then, translate the dilated circle 8 units to the \(\boldsymbol{right}\) and 9 units \(\boldsymbol{down}\).