Hi swerve,
So, there are a few ways you can approach this. I will show you two today. The first method is visual and the second is algebraic.
Visual method:
Advantages: quicker method, easy, no memorization required
Disadvantages: Might not be great for people who struggle with visual perception
1) Quite simply, quickly sketch the line with the equation y=x (passes through the origin and has a slope of +1)
2) Roughly draw the point (-1,-2) relative to the line. Since we know that when x = -1, y=-1 (for the line), the point (-1,-2) will be one unit below the line. Also, since we know that when y=-2, x =-2, we know that the point (-1,-2) is 1 unit to the right of the line.
3) A symmetrical point on the other side of the line would have to deviate from the line in the opposite direction, but with the same magnitude. Simply put, the point's x-coordinate must be 1 unit directly to the LEFT (-1,-2) and the y-coordinate must be 1 unit directly ABOVE (-2,-2), so we end up with ( [-1 - 1] , [-2+1]), which leaves (-2,-1).
4) Our answer is thus equivalent to OA A.
Algebra:
Advantages: Systematic approach (i.e. if you follow this process with any straight line, you will be able to solve)
Disadvantages: More time-consuming, more steps (and therefore more susceptible to silly mistakes)
Theory: For a point to be symmetrical to another about a line of symmetry, the point must be on the line that is perpendicular to the line of symmetry)
1) Determine the equation of the line along which the symmetrical point must lie:
for perpendicular lines, gradient 1 * gradient 2 = -1
gradient 1 = 1; therefore, gradient 2 = -1
y = -x +b and we know that (-1,-2) is on the line
-2 = -(-1) +b
-2 = 1+b
-3=b
y = -x -3
2) Determine the x coordinate where the two lines cross:
x = -x-3
2x = -3
x = -1.5
3) Our x coordinate for the point is (-1,-2), therefore the x coordinate is 0.5 units larger than the intersection point above. For the symmetrical point, the x coordinate must be 0.5 units SMALLER than the intersection point above. Therefore, the point is at x = -1.5 - 0.5 = -2
4) Find the corresponding y coordinate: y = -(-2) -3 = 2-3 = -1
5) Therefore, the point is (-2,-1), OA A.
