Question

1. (4 points) determine what type of symmetry, if any, each shape has. place an x in the appropriate column(s).

name	shape	line symmetry	rotational symmetry	point symmetry	no symmetry
scalene triangle	scalene triangle image	x
regular hexagon	regular hexagon image
isosceles trapezoid	isosceles trapezoid image

1. (1 point) how many lines of symmetry does this figure have?

figure image
number of lines of symmetry: ______

bonus question (+3 points)
what is the location of the image of p(-8,1) after a counterclockwise rotation of 90° about the point (-3,7)? show your work! attach your tracing paper if you used it.
coordinate grid image with p(-8,1) marked
a. (-1,-8)
b. (2,2)
c. (3,2)

Explanation:

14.

• Square:
• Line Symmetry: A square has 4 lines of symmetry (two diagonals and two lines through the mid - points of opposite sides).
• Rotational Symmetry: It has rotational symmetry of order 4 (rotation by \(90^{\circ},180^{\circ},270^{\circ},360^{\circ}\) about its center maps the square onto itself).
• Point Symmetry: A square has point symmetry. If we rotate it \(180^{\circ}\) about its center, it maps onto itself.
• Scalene Triangle:
• A scalene triangle has no line of symmetry (since all its sides and angles are of different measures), no rotational symmetry (except the \(360^{\circ}\) rotation which is trivial for all shapes), and no point symmetry.
• Regular Hexagon:
• Line Symmetry: A regular hexagon has 6 lines of symmetry (three through opposite vertices and three through the mid - points of opposite sides).
• Rotational Symmetry: It has rotational symmetry of order 6 (rotation by \(60^{\circ},120^{\circ},\cdots,360^{\circ}\) about its center maps the hexagon onto itself).
• Point Symmetry: A regular hexagon has point symmetry. Rotating it \(180^{\circ}\) about its center maps it onto itself.
• Isosceles Trapezoid:
• Line Symmetry: An isosceles trapezoid has 1 line of symmetry (the line passing through the mid - points of the non - parallel sides).
• It has no rotational symmetry (except \(360^{\circ}\) rotation) and no point symmetry.

The given figure is composed of three circles and an equilateral triangle. An equilateral triangle has 3 lines of symmetry. Each line of symmetry of the equilateral triangle is also a line of symmetry of the entire figure.

Step 1: Translate the point

Let the center of rotation be \(C(-3,7)\). First, we translate the point \(P(-8,1)\) so that the center of rotation \(C(-3,7)\) becomes the origin.
The translation rule is \((x,y)\to(x + 3,y - 7)\).
For point \(P(-8,1)\), the translated point \(P'\) has coordinates \((-8+3,1 - 7)=(-5,-6)\)

Step 2: Rotate the translated point

The rule for a counter - clockwise rotation of \(90^{\circ}\) about the origin \((x,y)\to(-y,x)\).
For the point \(P'(-5,-6)\), after rotation, the point \(P''\) has coordinates \((6,-5)\)

Step 3: Translate back

Now we translate the point \(P''(6,-5)\) back using the rule \((x,y)\to(x - 3,y + 7)\)
The coordinates of the final point \(P'''\) are \((6-3,-5 + 7)=(3,2)\)

Answer:

Name	Shape	Line Symmetry	Rotational Symmetry	Point Symmetry	No Symmetry
Scalene Triangle		X			X
Regular Hexagon		X	X	X
Isosceles Trapezoid		X

15.