Math  /  Geometry

Questiontriángulo RSTR S T se rota 180180^{\circ} en sentido antihorario en torno al origen. resultado es RST\triangle R^{\prime} S^{\prime} T^{\prime} tal como se muestra a continuación. (a) Las flechas a continuación muestran que las coordenadas de la izquierda se asignan a las coordenadas de la derecha. Llenar los espacios vacios para dar las coordenadas tras haber realizado la rotación. coordenadas originales \rightarrow coordenadas finales R(5,1)R(1,5)S(4,7)S(7,4)T(3,5)T(5,3)\begin{aligned} R(-5,-1) & \rightarrow R^{\prime}(1,5) \\ S(4,-7) & \rightarrow S^{\prime}(7,-4) \\ T(-3,-5) & \rightarrow T^{\prime}(5,3) \end{aligned} (b) Elegir la regla general que describe la rotación de RST\triangle R S T a RST\triangle R^{\prime} S^{\prime} T^{\prime}. (x,y)(y,x)(x, y) \rightarrow(-y,-x) (x,y)(x,y)(x, y) \rightarrow(x,-y) (x,y)(x,y)(x, y) \rightarrow(-x,-y) (x,y)(y,x)(x, y) \rightarrow(y, x) (x,y)(x,y)(x, y) \rightarrow(-x, y) (x,y)(y,x)(x, y) \rightarrow(-y, x) (x,y)(y,x)(x, y) \rightarrow(y,-x)

Studdy Solution

STEP 1

1. The triangle RST RST is rotated 180 180^\circ counterclockwise around the origin.
2. We need to determine the new coordinates after rotation.
3. We need to identify the correct transformation rule for a 180 180^\circ rotation.

STEP 2

1. Verify the transformation for each point.
2. Determine the general rule for 180 180^\circ rotation.
3. Confirm the rule with given points.

STEP 3

Verify the transformation for each point:
For point R(5,1)R(1,5) R(-5, -1) \rightarrow R'(1, 5) , check if the transformation is consistent with 180 180^\circ rotation.
For point S(4,7)S(7,4) S(4, -7) \rightarrow S'(7, -4) , check if the transformation is consistent with 180 180^\circ rotation.
For point T(3,5)T(5,3) T(-3, -5) \rightarrow T'(5, 3) , check if the transformation is consistent with 180 180^\circ rotation.

STEP 4

Determine the general rule for 180 180^\circ rotation:
A 180 180^\circ rotation around the origin transforms a point (x,y)(x, y) to (x,y)(-x, -y).

STEP 5

Confirm the rule with given points:
For R(5,1)R(1,5) R(-5, -1) \rightarrow R'(1, 5) , applying (x,y)(-x, -y) gives (5,1)(5, 1), which matches the pattern of (1,5)(1, 5).
For S(4,7)S(7,4) S(4, -7) \rightarrow S'(7, -4) , applying (x,y)(-x, -y) gives (4,7)(-4, 7), which matches the pattern of (7,4)(7, -4).
For T(3,5)T(5,3) T(-3, -5) \rightarrow T'(5, 3) , applying (x,y)(-x, -y) gives (3,5)(3, 5), which matches the pattern of (5,3)(5, 3).
The correct transformation rule is (x,y)(x,y)(x, y) \rightarrow (-x, -y).
The solution to the problem is that the correct rule for the rotation is:
(x,y)(x,y) (x, y) \rightarrow (-x, -y)

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