advertisement

TEKS: G2B, G10A, G5C The student will make conjectures about angles, lines and polygons using a variety of approaches including transformations. The student will use congruence transformations to make conjectures and justify properties of figures. The student will use prosperities of transformations to make connections between math and the real world, such as tessellations. A tessellation, or tiling, is a repeating pattern that completely covers a plane with no gaps or overlaps. The measures of the angles that meet at each vertex must add up to 360°. In the tessellation shown, each angle of the quadrilateral occurs once at each vertex. Because the angle measures of any quadrilateral add to 360°, any quadrilateral can be used to tessellate the plane. Four copies of the quadrilateral meet at each vertex. The angle measures of any triangle add up to 180°. This means that any triangle can be used to tessellate a plane. Six copies of the triangle meet at each vertex as shown. Example: 1 Reflect the rectangle with vertices A(-2, 4), B(3, 4), C(3, 3) and D(–2, 3) across the y=1 line. A B D C D’ C’ A’ B’ A(-2, 4) A (-2, –2) B(3, 4) B ’(3, –2) C(3, 3) C’(3, –1) D(–2, 3) D ’(–2, –1) Graph the image and preimage. Example: 2 Reflect the rectangle with vertices A-2, 4), B(3, 4), and C(–2, 3) across the x=-1 line. A’ A C B B’ C’ A(-5, 4) A’ (3, 4) B(-3, 2) B ’(1, 2) C(-5, 2) C’(3, 2) Graph the image and preimage. REVIEW of Rules Reflection: x axis (x, y) → (x, - y) y axis (x, y) → (- x, y) y = x (x, y) → (y, x) Rotation: 90° counterclockwise about the origin (x, y) → (- y, x) 180° counterclockwise about the origin (x, y) → (- x,- y) Translation: (x, y) → (x + a, y + b) Dilation: (x, y) → (kx, ky) REVIEW A transformation is a change in the position, size, or shape of a figure. The original figure is called the preimage. The resulting figure is called the image. A transformation maps the preimage to the image. Arrow notation () is used to describe a transformation, and primes (’) are used to label the image. Example: 3 Identify each transformation. Then use arrow notation to describe the transformation. a. translation; MNOP M’N’O’P’ b. rotation; ∆XYZ ∆X’Y’Z’ Example: 4 Identify the transformation. Reflection across the x axis, DEFG D’E’F’G’ Example: 5 Reflect the figure with the given vertices across the given line. X(2, –1), Y(–4, –3), Z(3, 2); y-axis The reflection of (x, y) is (-x, y). Z’ Z X’ Y X(2,–1) X’ (-2, -1) Y(–4,–3) Y’ (4, -3) Z(3, 2) X Y’ Z ’(-3, 2) Graph the image and preimage. Example: 6 Reflect the figure with the given vertices across the given line. R(–2, 2), S(5, 0), T(3, –1); y = x S’ R The reflection of (x, y) is (y, x). T’ S R’ T R(–2, 2) R’ (2, –2) S (5, 0) S’ (0, 5) T(3, –1) T ’(–1, 3) Graph the image and preimage. Example: 7 Reflect the rectangle with vertices S(3, 4), T(3, 1), U(–2, 1) and V(–2, 4) across the x-axis. V U U’ V’ S T T’ S’ The reflection of (x, y) is (x,–y). S(3, 4) S’ (3, –4) T(3, 1) T ’(3, –1) U(–2, 1) U ’(–2, –1) V(–2, 4) V ’(–2, –4) Graph the image and preimage. To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) (x + a, y + b). Example: 8 Find the coordinates for the image of ∆ABC after the translation (x, y) (x + 2, y - 1). Draw the pre image and image. Step 1 Find the coordinates of ∆ABC. The vertices of ∆ABC are A(–4, 2), B(–3, 4), C(–1, 1). Step 2 Apply the rule to find the vertices of the image. A’(–4 + 2, 2 – 1) = A’(–2, 1) B’(–3 + 2, 4 – 1) = B’(–1, 3) C’(–1 + 2, 1 – 1) = C’(1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. Example: 9 Find the coordinates for the image of JKLM after the translation J’ K’ (x, y) (x – 2, y + 4). Draw the image. Step 1 Find the coordinates of JKLM. The vertices of JKLM are J(1, 1), K(3, 1), L(3, –4), M(1, –4), . M’ Step 2 Apply the rule to find the vertices of the image. J’(1 – 2, 1 + 4) = J’(–1, 5) K’(3 – 2, 1 + 4) = K’(1, 5) L’(3 – 2, –4 + 4) = L’(1, 0) M’(1 – 2, –4 + 4) = M’(–1, 0) Step 3 Plot the points. Then finish drawing the image by using a straightedge to connect the vertices. L’ Example: 10 The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (2, –1) and A’ has A’ coordinates Step 2 Translate. To translate A to A’, 2 units are subtracted from the x-coordinate and 1 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x – 3, y + 1 ). A Example: 11 Use the diagram to write a rule for the translation of square 1 to square 3. Step 1 Choose two points. Choose a Point A on the preimage and a corresponding Point A’ on the image. A has coordinate (3, 1) and A’ has coordinates (–1, –3). Step 2 Translate. To translate A to A’, 4 units are subtracted from the x-coordinate and 4 units are subtracted from the y-coordinate. Therefore, the translation rule is (x, y) (x – 4, y – 4). A’ Counterclockwise Example: 12 Describe the transformation. 90° rotation counterclockwise, ∆ABC ∆A’B’C’ Example: 13 Rotate ∆ABC by 90° about the origin. B’ The rotation of (x, y) is (–y, x). A(2, –1) A’ (1, 2) B(4, 1) B’ (–1, 4) C(3, 3) C’ (–3, 3) C’ C A’ B A Graph the preimage and image. Example: 14 Rotate ΔJKL with vertices J(3, 5), K(4, –5), and L(–1, 6) by 180° about the origin. The rotation of (x, y) is (–x, –y). J(3, 5) K(4, –5) L(–1, 6) L K’ J J’ (–3, -5) K’ (–4, 5) L’ (1, –6) Graph the preimage and image. J’ L’ K A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) (ka, kb). Helpful Hint If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. The scale factor in fraction form is an improper fraction. If the scale factor is less than 1 (k < 1), it is a reduction. The scale factor in fraction form is a proper fraction. Example: 15 Draw the border of the photo after a dilation with scale factor Example: 15 cont. Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by Rectangle ABCD Rectangle A’B’C’D’ Example: 15 cont. Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0). Draw the rectangle. Example: 16 Given that ∆TUO ~ ∆RSO, find the coordinates of U and the scale factor. Since ∆TUO ~ ∆RSO, Substitute 12 for RO, 9 for TO, and 16 for OY. 12OU = 144 OU = 12 Cross Products Prop. Divide both sides by 12. Example: 16cont. U lies on the y-axis, so its x-coordinate is 0. Since OU = 12, its ycoordinate must be 12. The coordinates of U are (0, 12). So the scale factor is