image appears like the pre-image that has been moved through a - PowerPoint PPT Presentation

D AY 61 – T RIANGLE CONGRUENCE IN TERMS OF RIGID MOTIONS

I NTRODUCTION A rigid motion transforms a figure such that the shape and size remain the same. When two figures have the same size and shape, they are said to be congruent. In this lesson, we are going to summarize concepts on the effect of rigid motion on a given figure and to use the definition of congruence in terms of the rigid motion to decide whether two figures are congruent.

V OCABULARY  Congruent figures When there exists one or more rigid motions which can map two figures, then those figures are said to be congruent

There are four rigid motions each having an effect on a given figure. A figure that is transformed by a translation, the image appears like the pre-image that has been moved through a certain distance with the orientation remains unchanged. Consider the ∆ 𝐵𝐶𝐷 below C 𝐷′ A B 𝐵′ 𝐶′

In a reflection, the object distance and the image distance from the mirror line remain the same remains the same and the orientation changes to the opposite direction. 𝐷′ 𝐷 𝐵′ 𝐶 𝐵′ 𝐶′

In a glide reflection, the object is reflected then the image is moved in a certain distance along the mirror line. The orientation changes in the opposite direction. 𝐷 𝐷′ 𝐵′ 𝐶 𝐶′ 𝐵′

If there exists a rigid motion that can map two figures, then the figures are congruent. Given two figures, we need to identify a rigid motion which can map them, to decide whether they are congruent. If the rigid motion is identified then, the figures are congruent. Consider the triangles below. There is no rigid motion which can map these two triangles. Thus the two triangles are not congruent.

Example Using the definition of congruence in terms of rigid motion, state whether ∆𝑁𝑂𝑃 ≅ ∆𝑇𝑈𝑆 R 𝑃 T 𝑁 S 𝑂

Solution By the definition of congruence, we need to identify a rigid motion that will map ∆𝑁𝑂𝑃 onto ∆𝑇𝑈𝑆 . A reflection over a vertical line half way the distance between the two triangles will map ∆𝑁𝑂𝑃 onto ∆𝑇𝑈𝑆 . Thus, ∆𝑁𝑂𝑃 ≅ ∆𝑇𝑈𝑆

HOMEWORK Is ∆𝐵𝐶𝐷 congruent to ∆𝐾𝐿𝑀? Why? C L A J K B

A NSWERS TO HOMEWORK Yes Because the figures are mapped onto one another by a translation which is a rigid motion.

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