Answer:
Step-by-step explanation:
To find the coordinates of point S, which is the symmetric point of point R with respect to the line y = x, we can use the formula for finding the reflection of a point across a line.
First, let's understand what it means for a point to be symmetric to another point with respect to a line. In this case, point S is the reflection of point R across the line y = x. This means that the line connecting point R and point S is perpendicular to the line y = x, and the distance between point R and the line y = x is the same as the distance between point S and the line y = x.
To find the coordinates of point S, we can use the following steps:
1. Determine the slope of the line y = x. The slope of a line with equation y = mx + b is equal to the coefficient of x, which is 1 in this case.
2. Determine the equation of the line passing through point R with slope -1. Since the line y = x is perpendicular to the line connecting point R and point S, the slope of this line will be the negative reciprocal of the slope of y = x. Therefore, the equation of the line passing through point R is y = -x + b.
3. Substitute the coordinates of point R (2, -5) into the equation y = -x + b to solve for the value of b. In this case, we have -5 = -(2) + b. Solving for b, we get b = -5 + 2 = -3.
4. Substitute the value of b into the equation y = -x + b to obtain the equation of the line passing through point R. In this case, the equation of the line passing through point R is y = -x - 3.
5. Find the intersection point of the line y = -x - 3 and the line y = x. To do this, we can set the two equations equal to each other and solve for x. In this case, we have -x - 3 = x. Solving for x, we get x = -1.
6. Substitute the value of x = -1 into either equation to find the value of y. In this case, we can substitute x = -1 into the equation y = -x - 3. Solving for y, we get y = -(-1) - 3 = -1 - 3 = -4.
Therefore, the coordinates of point S are (-1, -4).