Day 69 – Similarity of two figures

Introduction We have done quite a number of lessons that talks about identical figures. At times, our interest may be slightly different. We would be provided with a model of an item then asked to mold the real items based on the model. This is a common practice is building and construction where architectural drawings (plans and elevations) are used by contractors to construct a house. The two items, the plans and the real house has some relation that we would like discuss. In this lesson, we will decide if two given images are similar or not.

Vocabulary 1. similarity transformations These are two sets of transformations, one or more rigid transformation following by a dilation 2. Rigid transformation A transformation that mains the size and the angular measure of the object being transformed 3. Similarity Is a term describing two or more figures whose corresponding angles are equal and corresponding sides are proportional

Similarity Two images are similar (i). Corresponding angles are equal (ii). Corresponding sides are similar. This implies that the ratio between the corresponding sides (what we call a linear scale factor) is the same for all the pairs of sides taken. If we have two triangles ABC and EFG, then the two are similar if (i)∠𝐴=∠𝐸,∠𝐵=∠𝐹 and ∠𝐶=∠𝐺 (ii) 𝐴𝐵 𝐸𝐹 = 𝐵𝐶 𝐹𝐺 = 𝐶𝐴 𝐺𝐸 =𝑘 where 𝑘 is called the linear scale factor.

A linear scale factor exists if the sides are dilated and angular measure maintained. This implies that, we must have one or more rigid motion then a dilation for two images to be similar. Example Are the two figures similar? S B F T U Y 4 in 5 in 3 in 4.5 in 7.5 in 6 in

The two figures have the same orientation and shape hence the corresponding angles are equal. Since they one is a distance from another, it implies translation was done. Let FSB be the pre-image and YTU an image. Then if 𝑘 is the dilation factor, we have 𝑘𝐹𝑆=𝑌𝑇, 𝑘𝑆𝐵=𝑇𝑈 and 𝑘𝐵𝐹=𝑈𝑌. 3𝑘=4.5𝑖𝑛, thus 𝑘= 4.5 3 =1.5 4𝑘=4×1.5=6 𝑖𝑛 5𝑘=5×1.5=7.5𝑖𝑛 Since there is a translation and the dilation of scale factor 1.5, the two images are similar.

homework In the figure below, ∠𝑌𝐷𝑅 is a right angle. U and P are the midpoints of ER and ET respectively. Find the linear scale factor between triangle EUP and ERT. R T E U P

Answers to homework 0.5

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