Day 84 – Enlargement

Introduction If we cut a square through a line joining the midpoints of its sides, we will get four small squares of the same size. These squares will have the same shape as the original square but with a different size. To get the original square, we need to proportionally increase the length of the sides of one of the small squares. In this lesson, we discuss enlargement of such figures.

Vocabulary Center of Enlargement This is a point where the lines joining the points of the image with the corresponding points of the object converge. Scale Factor This is the ratio of image distance to the object/pre- image distance with respect to the center of enlargement.

When two figures are such that the first one is smaller than the second one and they have the same shape, then the latter is an enlargement of the former. Consider the diagram below. 𝐵′ 𝐴 𝐴′ 𝐵 𝐶′ 𝐶 O

∆𝐴′𝐵′𝐶′ is an enlargement of ∆𝐴𝐵𝐶 Point O is called the center of enlargement. The length from point O to each point on the image is proportional to the length from point O to the corresponding point on the pre-image. 𝑂 𝐴 ′ 𝑂𝐴 = 𝑂 𝐵 ′ 𝑂𝐵 = 𝑂 𝐶 ′ 𝑂𝐶 This ratio is the linear scale factor of enlargement. This scale factor can also be found by determining the ratio of the length of one side of the image to the length of the corresponding side of the pre-image. 𝐴 ′ 𝐵 ′ 𝐴𝐵 = 𝐴 ′ 𝐶 ′ 𝐴𝐶 = 𝐵 ′ 𝐶 ′ 𝐵𝐶 = 𝑂 𝐴 ′ 𝑂𝐴 = 𝑂 𝐵 ′ 𝑂𝐵 = 𝑂 𝐶 ′ 𝑂𝐶 =𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 The image is always similar to the object under enlargement.

When the linear scale factor is 1, the size of the object is not changed. The image and the pre-image are congruent. The image coincides with the pre-image if the center of enlargement lies on the pre-image. When the linear scale factor is -1, the size of the object is also not changed but the image is inverted.

Enlargement on coordinate plane Enlargement can also be done a coordinate plane. If the center of enlargement is the origin, we multiply the coordinates of the object with the scale factor to get the coordinates of the image. For instance if we want to enlarge a triangle with the vertices at points 𝐴 1,1 , 𝐵(3,1) and 𝐶 1,3 with a scale factor of 2, we multiply these coordinates with 2 to get the coordinates of the image as 𝐴′ 2,2 , 𝐵′(6,2) and 𝐶′ 2,6 . Consider this triangle and its image in the following grid.

. -4 -2 0 2 4 6 8 𝒙 𝟐 𝟒 𝟔 −𝟐 𝒚 𝑨 𝑩 𝑪 𝑨′ 𝑪′ 𝑩′

Example What is the scale factor of enlargement in the diagram below Example What is the scale factor of enlargement in the diagram below. Solution Scale factor = 15 6 =2.5 15 𝑖𝑛 6 𝑖𝑛 8 𝑖𝑛

homework A triangle has its vertices at points 𝐽 −2,2 , 𝐾 2,3 and 𝐿(2,−2). Without drawing find the coordinates of the image if it is enlarged with a scale factor of 3 about the origin.

Answers to homework 𝐽′ −6,6 , 𝐾′ 6,9 and 𝐿′(6,−6)

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