Question

in exercise 10, determine whether the figure has rotational symmetry. if so, describe any rotations that map the figure onto itself.
10.
check one yes____ no____
degrees of rotation______
in exercises 11 and 12, find the scale factor of the dilation. then tell whether the dilation is a reduction or an enlargement.
11.
12.
scale factor______
circle one: reduction enlargement
scale factor______
circle one: reduction enlargement
in exercises 13 and 14, given the polygon with the given vertices write the coordinates of its image after a dilation with scale factor k.

1. ( p(1,2) ), ( q(2,2) ), ( r(4,-2) ), ( s(-1,-3) ); ( k = 4 )

( p(quad,quad) quad q(quad,quad) quad r(quad,quad) quad s(quad,quad) )

1. ( a(-4,4) ), ( b(-2,6) ), ( c(6,-6) ), ( d(-2,-4) ); ( k=\frac{1}{2} )

( a(quad,quad) quad b(quad,quad) quad c(quad,quad) quad d(quad,quad) )

Explanation:

Step1: Determine rotational symmetry for exercise 10

Rotational symmetry means a figure maps onto itself after a rotation.
For the given figure, it has rotational symmetry.
The degrees of rotation: \(90^{\circ},180^{\circ},270^{\circ}\) (since it is a four - pointed star - like shape with a central circle, a rotation of \(90^{\circ}\) around the center will map it onto itself).
So, check “Yes” and degrees of rotation are \(90^{\circ},180^{\circ},270^{\circ}\)

Step2: Find scale factor for exercise 11

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of image segment}}{\text{length of pre - image segment}}\)
Here, \(k = \frac{1.5}{9}=\frac{1}{6}\)
Since \(k=\frac{1}{6}<1\), it is a reduction

Step3: Find scale factor for exercise 12

\(k=\frac{12}{8} = 1.5\)
Since \(k = 1.5>1\), it is an enlargement

Step4: Find coordinates for exercise 13

For a dilation with scale factor \(k\), if a point \(P(x,y)\) is dilated, the image \(P^{\prime}(kx,ky)\)
For \(P(1,2)\), \(P^{\prime}(1\times4,2\times4)=(4,8)\)
For \(Q(2,2)\), \(Q^{\prime}(2\times4,2\times4)=(8,8)\)
For \(R(4, - 2)\), \(R^{\prime}(4\times4,-2\times4)=(16,-8)\)
For \(S(-1,-3)\), \(S^{\prime}(-1\times4,-3\times4)=(-4,-12)\)

Step5: Find coordinates for exercise 14

For \(A(-4,4)\), \(A^{\prime}(-4\times\frac{1}{2},4\times\frac{1}{2})=(-2,2)\)
For \(B(-2,6)\), \(B^{\prime}(-2\times\frac{1}{2},6\times\frac{1}{2})=(-1,3)\)
For \(C(6,-6)\), \(C^{\prime}(6\times\frac{1}{2},-6\times\frac{1}{2})=(3,-3)\)
For \(D(-2,-4)\), \(D^{\prime}(-2\times\frac{1}{2},-4\times\frac{1}{2})=(-1,-2)\)

Answer:

1. Check one: Yes. Degrees of rotation: \(90^{\circ},180^{\circ},270^{\circ}\)
2. Scale Factor: \(\frac{1}{6}\). Circle one: Reduction
3. Scale Factor: \(1.5\). Circle one: Enlargement
4. \(P^{\prime}(4,8)\), \(Q^{\prime}(8,8)\), \(R^{\prime}(16,-8)\), \(S^{\prime}(-4,-12)\)
5. \(A^{\prime}(-2,2)\), \(B^{\prime}(-1,3)\), \(C^{\prime}(3,-3)\), \(D^{\prime}(-1,-2)\)