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A point P (a, b) is reflected in the x-a...

A point P (a, b) is reflected in the x-axis to P'(2, -3). Write down the values of a and b. P" is the image of P, reflected in the y-axis. Write down the co-ordinates of P". Find the co-ordinates of P", when P is reflected in the line, parallel to y-axis, such that x = 4.

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To solve the problem step by step, we will follow the instructions given in the question regarding the reflections of point P in the x-axis, y-axis, and a line parallel to the y-axis (x = 4). ### Step 1: Find the values of a and b We know that point P(a, b) is reflected in the x-axis to P'(2, -3). 1. When a point (x, y) is reflected in the x-axis, the x-coordinate remains the same, and the y-coordinate changes sign. Thus, the reflection of P(a, b) in the x-axis is P'(a, -b). 2. According to the problem, P' = (2, -3). Therefore, we can set up the following equations: - a = 2 - -b = -3 From the second equation, we can deduce that: - b = 3 Thus, we find: - a = 2 - b = 3 ### Step 2: Write down the coordinates of P The coordinates of point P are: - P(2, 3) ### Step 3: Find the coordinates of P'' when reflected in the y-axis Next, we need to find the coordinates of P'' when point P is reflected in the y-axis. 1. When a point (x, y) is reflected in the y-axis, the y-coordinate remains the same, and the x-coordinate changes sign. Thus, the reflection of P(2, 3) in the y-axis is P''(-a, b). 2. Substituting the values of a and b: - P'' = (-2, 3) ### Step 4: Find the coordinates of P''' when reflected in the line x = 4 Now we need to find the coordinates of P''' when point P is reflected in the line x = 4. 1. The line x = 4 is a vertical line. To find the reflection of a point across a vertical line, we calculate the distance from the point to the line and then move the same distance on the opposite side of the line. 2. The x-coordinate of point P is 2. The distance from P(2, 3) to the line x = 4 is: - Distance = 4 - 2 = 2 3. To find the coordinates of the reflected point P''', we move 2 units to the right of the line x = 4: - New x-coordinate = 4 + 2 = 6 - The y-coordinate remains the same, so it is still 3. Thus, the coordinates of P''' are: - P'''(6, 3) ### Summary of Results - Values of a and b: a = 2, b = 3 - Coordinates of P: P(2, 3) - Coordinates of P'' after reflection in the y-axis: P''(-2, 3) - Coordinates of P''' after reflection in the line x = 4: P'''(6, 3)
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