Day 54 – Congruence in opposite dilation

Introduction When an object and the image are on the opposite sides of the center of dilation and the image is inverted, then the scale factor is a negative number. If this scale factor is -1, the object and the image have the same size and shape. In this lesson we will show congruence of image distance and object distance if the scale factor is -1.

Vocabulary Opposite dilation This is when the image and the object are on the opposite sides of the center of dilation.

Consider the illustration below where ∆𝐴𝐵𝐶 is dilated with a scale factor of -1 to form an image on the opposite side of the center of dilation O. The distances, 𝑂𝐴=𝑂 𝐴 ′ =1𝑖𝑛 𝑂𝐵=𝑂 𝐵 ′ =3𝑖𝑛 𝑂𝐶=𝑂 𝐶 ′ =2.5𝑖𝑛 O A B C 𝐶′ 𝐵′ 𝐴′

Note that two lines are congruent if they have the same length Note that two lines are congruent if they have the same length. Thus 𝑂𝐴≅𝑂 𝐴 ′ 𝑂𝐵≅𝑂 𝐵 ′ 𝑂𝐶≅𝑂 𝐶 ′ ∆𝐴𝐵𝐶 is also congruent to its image since the angle measure remains the same and the corresponding sides are equal. The image is inverted and has the same size and shape as the object. The image has opposite orientation.

Let us consider dilation in coordinate plane with a scale factor of -1. -6 -4 -2 0 2 4 6 x 4 2 -2 -4 y 𝐴 𝐵 𝐶

The coordinates of the object are 𝐴 1,1 , 𝐵 1,4 and 𝐶(4,4) The coordinates of the object are 𝐴 1,1 , 𝐵 1,4 and 𝐶(4,4). Since the scale factor is -1, we multiply this coordinates by -1 to get the coordinates of the image as 𝐴′ −1,−1 , 𝐵′ −1,−4 and 𝐶′(−4,−4).

. -6 -4 -2 0 2 4 6 x 4 2 -2 -4 y 𝐶′ 𝐵′ 𝐴′ 𝐴 𝐵 𝐶

𝑂𝐴=𝑂 𝐴 ′ 𝑂𝐵=𝑂 𝐵 ′ 𝑂𝐶=𝑂 𝐶 ′ Thus 𝑂𝐴≅𝑂 𝐴 ′ 𝑂𝐵≅𝑂 𝐵 ′ 𝑂𝐶≅𝑂 𝐶 ′ ∆𝐴𝐵𝐶 is also congruent to its image since the angle measure remains the same and the corresponding sides are equal. The image is inverted and has the same size and shape as the object.

Example The triangle below is dilated with a scale factor of -1 Example The triangle below is dilated with a scale factor of -1. Indicate the image if point O is the center of dilation. O

Solution O

homework Rectangle ABCD below is dilated with a scale factor of -1 about the point B. If the triangle measures 4 in by 3in, what is the distance from 𝐵′ 𝐷′ of the image 𝐴 ′ 𝐵 ′ 𝐶 ′ 𝐷′ ? A B D C

Answers to homework 5 in

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