Answer to Question #99351 in Analytic Geometry for Gloria

Question

The vertices of quadrilateral OPQR are O(0,0), P(2,0), Q(4,2), R(0,3). The vertices of its image under a rotation are O'(1,-1), P'(1,-3), Q'(3,-5) and R'(4,-1).
(a)(i) On the grid draw OPQR and its image O'P'Q'R'.
(ii) by construction determine the centre and angle of rotation.
(b) On the same grid as (a) (i) above, draw O''P''Q''R'', the image of O'P'Q'R' under a reflection in the line y = x
(c)From the quadrilaterals drawn, state the pairs that are:
(i) Directly congruent; (
(ii) Oppositely congruent

Answer 1

$$(x_O-x_C)^2+(y_O-y_C)^2=(x_{O'}-x_C)^2+(y_{O'}-y_C)^2$$ $$x_C^2+y_C^2=1-2x_C+x_C^2+1+2y_C+y_C^2$$ $$y_C=x_C-1$$

$$(x_P-x_C)^2+(y_P-y_C)^2=(x_{P'}-x_C)^2+(y_{P'}-y_C)^2$$ $$(2-x_C)^2+y_C^2=(1-x_C)^2+(-3-y_C)^2$$ $$4-4x_C+x_C^2+y_C^2=1-2x_C+x_C^2+9+6y_C+y_C^2$$ $$-2x_C=6+6y_C$$

$$y_C=x_C-1$$ $$x_C=-3-3y_C$$

$$x_C=0, y_C=-1$$

Arbitrary Rotation Center

To rotate about an arbitrary point $C(0,-1)$ by $\theta:$

Translate the object so that $C$ will coincide with the origin:

$$O_1(0, 1), P_1(2,1), Q_1(4, 3),R_1(0,4)$$

Rotate the object:

$$O_2(0\cdot\cos\theta-1\cdot\sin\theta,0\cdot\sin\theta+1\cdot\cos\theta),$$ $$P_2(2\cdot\cos\theta-1\cdot\sin\theta,2\cdot\sin\theta+1\cdot\cos\theta),$$ $$Q_2(4\cdot\cos\theta-3\cdot\sin\theta,4\cdot\sin\theta+3\cdot\cos\theta),$$ $$R_2(0\cdot\cos\theta-4\cdot\sin\theta,0\cdot\sin\theta+4\cdot\cos\theta),$$

Translate the object back:

$$O'(-\sin\theta,\cos\theta-1),$$ $$P'(2\cos\theta-\sin\theta,2\sin\theta+\cos\theta-1),$$ $$Q'(4\cos\theta-3\sin\theta,4\sin\theta+3\cos\theta-1),$$ $$R'(-4\sin\theta,4\cos\theta-1),$$

$$-\sin\theta=1$$ $$\cos\theta-1=-1$$

$$2\cos\theta-\sin\theta=1$$ $$2\sin\theta+\cos\theta-1=-3$$

$$4\cdot\cos\theta-3\cdot\sin\theta=3$$ $$4\sin\theta+3\cos\theta-1=-5$$

$$-4\sin\theta=4$$ $$4\cos\theta-1=-1$$

$$\theta=270\degree$$ $$x_C=0$$ $$y_C=-1$$

(c)From the quadrilaterals drawn, state the pairs that are:

(i) Directly congruent

When figures are congruent and have the same orientation. For example, the image and pre image in a rotation and translation.

$$OPQR \ and\ O'P'Q'R'$$

(ii) Oppositely congruent

When figures are congruent but have opposite orientations. For example, the image and pre image in a reflection and glide reflection.

$$OPQR \ and\ O''P''Q''R''$$

$$O'P'Q'R' \ and\ O''P''Q''R''$$