Such a transformation is called a rotation. You do this by rotating (turning) one shape to align with the other. To see if the two triangles are similar, you first have to get them both in the same direction, or orientation. The scalene triangle on the left and the scalene triangle on the right are actually similar, but the one on the right has been rotated to stand on its shortest side. You use all of these concepts in everyday life. Translation - Shapes are slid across the plane.Reflection - Shapes are flipped across an imaginary line to make mirror images.Rotation - Shapes are rotated or turned around an axis.Like restricted game pieces on a game board, you can move two-dimensional shapes in only three ways: Geometry transformations are movements of two-dimensional shapes in two dimensions, or within their plane. Two equilateral triangles, each with one side 90 meters long, are congruent. Polygons (or any geometric shapes) are congruent if they are the same size and shape. Yes the proportions of the two isosceles triangles are the same, so the two triangles are similar. If the ratio of one side and one leg of the left-hand triangle is the same ratio as the corresponding side and leg of the right-hand triangle, they are proportional to each other, so they are similar. The right triangle has 30 cm legs and a 20 cm third side. Notice the left triangle has two legs 15 cm long and a third side, 10 cm long. Recall that the equal sides of an isosceles triangle are called legs. Next, you have to compare corresponding sides to see if they maintain the same ratio. You check and the corresponding angles between legs and third sides are congruent, at 71°. Are they similar? You have to check their interior angles to see if they are the same in both isosceles triangles. Or like your dog Bailey and the neighborhood dog Buddy.Ĭongruent objects are also similar, but similar objects are not congruent.īelow are two isosceles triangles, one with sides twice as long as the other. A shoe box for a size 4 child's shoe may be similar to, but smaller than, a shoe box for a man's size 14 shoe. Two geometric shapes are similar if they have the same shape but are different in size. Our example may sound a bit silly, but in geometry we use transformations all the time to bring two objects near each other, turn them to face the same way, and, if necessary, flip them to see if they are similar. You would have to wake Bailey up and get the two dogs facing the same direction, so you could compare snouts, and ears, and tails. You could bring Bailey and Buddy together. You are interested in seeing Bailey and Buddy side by side to compare them, but your dog is curled up, asleep on the couch, and the other dog is down the block. Imagine you are told that a small dog, Buddy, in your neighborhood looks exactly like your own big dog Bailey. neither with the results of quadratic equation nor with the intervals of monotony(sign) of the derivative.You love your dog Bailey. And it is decreasing on ( − ∞, − 1 − 45 2 ), then increasing on ( − 1 − 45 2, − 3 ) then decreasing on (-3,2) then increasing again on ( 2, ∞ ). But Instead the solutions of the third equation are x 1 = − 1 − 5 2 and x 2 = − 1 + 5 2. Therefore our function should increase on the interval ( − ∞, − 1 − 45 2 ) then decrease on the interval ( − 1 − 45 2, − 1 + 45 2 ) then increase again on the interval ( − 1 + 45 2, − 1 − i 3 2 ) and decrease again on ( − 1 + i 3 2, ∞ ). If you'll do the sign of the first one it'll be x 2 + x − 11 0 therefore x 1 = − 1 + 45 2 and x 1 = − 1 − 45 2, the last one x 2 + x + 1 > 0 has the solutions x 1 = − 1 − i 3 2 and x 2 = − 1 + i 3 2 (here I think that I did a mistake). The derivative is defined in the following order:ĭ d x f ( x ) = − x 2 − x + 11 ( x + 3 ) 2 e 2 − x for x 2. Some doubts with the sign of a derivative

Sequence of transformations