Lesson 15

Symmetry

  • Let’s describe some symmetries of shapes.

Problem 1

For each figure, identify any lines of symmetry the figure has.

A yin yang symbol.
A flag. A blue background with a white X.
A triskelion image, three congruent legs radiating from a center point.

 

Problem 2

In quadrilateral \(BADC\), \(AB=AD\) and \(BC=DC\). The line \(AC\) is a line of symmetry for this quadrilateral.

Quadrilateral B A D C.
  1. Based on the line of symmetry, explain why the diagonals \(AC\) and \(BD\) are perpendicular.
  2. Based on the line of symmetry, explain why angles \(ABC\) and \(ADC\) have the same measure.

Problem 3

Three line segments form the letter Z. Rotate the letter Z counterclockwise around the midpoint of segment \(BC\) by 180 degrees. Describe the result.

The letter Z, formed by 3 line segments. 4 points on the endpoints of the segments. Starting at the top left, A. Moving to the right, B. Slanting down and to the left, C. Moving to the right, D.
(From Unit 1, Lesson 14.)

Problem 4

There is a square, \(ABCS\), inscribed in a circle with center \(D\). What is the smallest angle we can rotate around \(D\) so that the image of \(A\) is \(B\)?

A:

\(45^\circ\)

B:

\(60^\circ\)

C:

\(90^\circ\)

D:

\(180^\circ\)

(From Unit 1, Lesson 14.)

Problem 5

Points \(A\), \(B\), \(C\), and \(D\) are vertices of a square.  Point \(E\) is inside the square. Explain how to tell whether point \(E\) is closer to \(A\), \(B\), \(C\), or \(D\).

(From Unit 1, Lesson 9.)

Problem 6

Lines \(\ell\) and \(m\) are perpendicular.

Sometimes reflecting a point over \(m\) has the same effect as rotating the point 180 degrees using center \(P\). Select all labeled points which have the same image for both transformations.

\(m \perp \ell\)

Perpendicular lines L and M.
A:

A

B:

B

C:

C

D:

D

E:

E

(From Unit 1, Lesson 11.)

Problem 7

Here is triangle \(POG\). Match the description of the rotation with the image of \(POG\) under that rotation.

Triangle P O G on isometric grid. P O and O G are both 2 units long. O G is vertical.
A:

Rotate 60 degrees clockwise around \(O\).

B:

Rotate 120 degrees clockwise around \(O\).

C:

Rotate 60 degrees counterclockwise around \(O\).

D:

Rotate 60 degrees clockwise around \(P\).

1:
Triangle P prime O prime G prime.
2:
Triangle P prime O prime G prime.
3:
Triangle P prime O prime G prime on an isometric grid. Point O prime faces upwards, P prime faces to the right, G prime faces to the left.
4:
Triangle P prime O prime G prime on isometric grid. P prime G prime is horizontal.
(From Unit 1, Lesson 13.)