Introduction Rigid motions can also be called congruency

Introduction Rigid motions can also be called congruency transformations. A congruency transformation moves a geometric figure but keeps the same size and shape. Preimages and images that are congruent are also said to be isometries. If a figure has undergone a rigid motion or a set of rigid motions, the preimage and image are congruent. When two figures are congruent, they have the same shape and size. Remember that rigid motions are translations, reflections, and rotations. Non-rigid motions are dilations, stretches, and compressions. Nonrigid motions are transformations done to a figure that change the figure’s shape and/or size. 1. 4. 2: Defining Congruence in Terms of Rigid Motions 1

Key Concepts • To decide if two figures are congruent, determine if the original figure has undergone a rigid motion or set of rigid motions. • If the figure has undergone only rigid motions (translations, reflections, or rotations), then the figures are congruent. • If the figure has undergone any non-rigid motions (dilations, stretches, or compressions), then the figures are not congruent. A dilation uses a center point and a scale factor to either enlarge or reduce the figure. A dilation in which the figure becomes smaller can also be called a compression. 1. 4. 2: Defining Congruence in Terms of Rigid Motions 2

Key Concepts, continued • A scale factor is a multiple of the lengths of the sides from one figure to the dilated figure. The scale factor remains constant in a dilation. • If the scale factor is larger than 1, then the figure is enlarged. • If the scale factor is between 0 and 1, then the figure is reduced. 3 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued • To calculate the scale factor, divide the length of the sides of the image by the lengths of the sides of the preimage. • A vertical stretch or compression preserves the horizontal distance of a figure, but changes the vertical distance. • A horizontal stretch or compression preserves the vertical distance of a figure, but changes the horizontal distance. 4 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued • To verify if a figure has undergone a non-rigid motion, compare the lengths of the sides of the figure. If the sides remain congruent, only rigid motions have been performed. • If the side lengths of a figure have changed, non-rigid motions have occurred. 5 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued Non-Rigid Motions: Dilations Enlargement/reduction Compare with. The size of each side changes by a constant scale factor. The angle measures have stayed the same. 6 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued Non-Rigid Motions: Vertical Transformations Stretch/compression Compare with. The vertical distance changes by a scale factor. The horizontal distance remains the same. Two of the angles have changed measures. 7 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Key Concepts, continued Non-Rigid Motions: Horizontal Transformations Stretch/compression Compare with. The horizontal distance changes by a scale factor. The vertical distance remains the same. Two of the angles have changed measures. 8 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Common Errors/Misconceptions • mistaking a non-rigid motion for a rigid motion and vice versa • not recognizing that rigid motions preserve shape and size • not recognizing that it takes only one non-rigid motion to render two figures not congruent 9 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice Example 1 Determine if the two figures to the right are congruent by identifying the transformations that have taken place. 10 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued 1. Determine the lengths of the sides. For the horizontal and vertical legs, count the number of units for the length. For the hypotenuse, use the Pythagorean Theorem, a 2 + b 2 = c 2, for which a and b are the legs and c is the hypotenuse. 11 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued AC = 3 CB = 5 AC 2 + CB 2 = AB 2 32 + 52 = AB 2 A′C′= 3 C′B′= 5 C′ A′ 2 + C′B′ 2 = A′B′ 2 32 + 52 = A′B′ 2 34 = AB 2 34 = A′B′ 2 The sides in the first triangle are congruent to the sides of the second triangle. 1. 4. 2: Defining Congruence in Terms of Rigid Motions 12

Guided Practice: Example 1, continued Note: When taking the square root of both sides of the equation, reject the negative value since the value is a distance and distance can only be positive. 13 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued 2. Identify the transformations that have occurred. The orientation has changed, indicating a rotation or a reflection. The second triangle is a mirror image of the first, but translated to the right 4 units. The triangle has undergone rigid motions: reflection and translation (shown on the next slide). 14 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued 15 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued 3. State the conclusion. The triangle has undergone two rigid motions: reflection and translation. Rigid motions preserve size and shape. The triangles are congruent. ✔ 16 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 1, continued 17 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice Example 2 Determine if the two figures to the right are congruent by identifying the transformations that have taken place. 18 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued 1. Determine the lengths of the sides. For the horizontal and vertical legs, count the number of units for the length. For the hypotenuse, use the Pythagorean Theorem, a 2 + b 2 = c 2, for which a and b are the legs and c is the hypotenuse. 19 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued AB = 3 AC = 4 AB 2 + AC 2 = CB 2 32 + 42 = CB 2 A′B′= 6 A′C′= 8 A′B′ 2 + A′C′ 2 = C′B′ 2 62 + 82 = C′B′ 2 25 = CB 2 100 = C′B′ 2 CB = 5 C′B′= 10 20 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued The sides in the first triangle are not congruent to the sides of the second triangle. They are not the same size. 21 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued 2. Identify the transformations that have occurred. The orientation has stayed the same, indicating translation, dilation, stretching, or compression. The vertical and horizontal distances have changed. This could indicate a dilation. 22 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued 3. Calculate the scale factor of the changes in the side lengths. Divide the image side lengths by the preimage side lengths. The scale factor is constant between each pair of sides in the preimage and image. The scale factor is 2, indicating a dilation. Since the scale factor is greater than 1, this is an enlargement. 1. 4. 2: Defining Congruence in Terms of Rigid Motions 23

Guided Practice: Example 2, continued 24 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued 4. State the conclusion. The triangle has undergone at least one non-rigid motion: a dilation. Specifically, the dilation is an enlargement with a scale factor of 2. The triangles are not congruent because dilation does not preserve the size of the original triangle. ✔ 25 1. 4. 2: Defining Congruence in Terms of Rigid Motions

Guided Practice: Example 2, continued 26 1. 4. 2: Defining Congruence in Terms of Rigid Motions