Question

7.determine whether the figure has line symmetry. if so, draw the line(s) of symmetry and write the number of lines of symmetry.
lines of symmetry
in exercises 8 and 9, write the coordinates of the image the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin.

1. ( j(2,4) ), ( k(3,1) ), ( l(1,-3) ), ( m(-1,2) ); ( 270^{circ} )

( 270^{circ} ) rule
( j(quad,quad) ) ( k(quad,quad) ) ( l(quad,quad) ) ( m(quad,quad) )

1. ( p(-2,4) ), ( q(3,1) ), ( r(-3,1) ), ( s(-4,2) ); ( 180^{circ} )

( 180^{circ} ) rule
( p(quad,quad) ) ( q(quad,quad) ) ( r(quad,quad) ) ( s(quad,quad) )

Explanation:

7.

Step1: Recall the definition of line symmetry

A line of symmetry is a line that divides a figure into two congruent parts.

Step2: Analyze the given figure

By visual inspection and using the concept of line - symmetry (folding the figure along a line and checking for congruence), we can find the number of lines of symmetry.
The figure has 4 lines of symmetry.

Step1: Recall the rotation rule for \(270^{\circ}\) counter - clockwise about the origin

The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y, - x)\).

Step2: Apply the rule to each vertex

• For \(J(2,4)\):

Using the rule \((x = 2,y = 4)\to J'(4,-2)\)

• For \(K(3,1)\):

Using the rule \((x = 3,y = 1)\to K'(1,-3)\)

• For \(L(1,-3)\):

Using the rule \((x = 1,y=-3)\to L'(-3,-1)\)

• For \(M(-1,2)\):

Using the rule \((x=-1,y = 2)\to M'(2,1)\)

Step1: Recall the rotation rule for \(180^{\circ}\) about the origin

The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\)

Step2: Apply the rule to each vertex

• For \(P(-2,4)\):

Using the rule \((x=-2,y = 4)\to P'(2,-4)\)

• For \(Q(3,1)\):

Using the rule \((x = 3,y = 1)\to Q'(-3,-1)\)

• For \(R(-3,1)\):

Using the rule \((x=-3,y = 1)\to R'(3,-1)\)

• For \(S(-4,2)\):

Using the rule \((x=-4,y = 2)\to S'(4,-2)\)

Answer:

4

8.