Question

quadrilateral abcd is symmetric with respect to the y axis. the coordinates of point a are (-2,2), and the coordinates of point c are (2, 1). if b is in the first quadrant, what are the coordinates of b?​

Answer 1

The coordinate of b = (2, 2)

Explanation:

The details of the quadrilateral abcd are;

The quadrilateral is symmetric about the y-axis (the line x = 0)

The coordinates of the vertices are; a(-2, 2), c(2, 1)

The location of the vertex, b = The first quadrant

We have;

The line
`\overline {ab}` is perpendicular to the line
`\overline {bc}`

Let (x, y) represent the coordinate of the vertex b, we have;

(y - 2)/(x - (-2)) = (y - 2)/(x + 2) = -1/(y - 1)/(x - 2) = - (x - 2)/(y - 1)

(y - 2)·(y - 1) = -(x + 2)·(x - 2)

y² + x² - 3·y - 2 = 0...(1)

y² + x² = 3·y + 2

Also we have;

(y - 2)² + (x - 1)² + (y - 2)² + (x - (-2))² = (2 - 1)² + (2 - (-2))² = 17

Therefore;

2·y² + 2·x² - 8·y + 2·x + 13 = 17

2·(3·y + 2) - 8·y + 2·x + 13 = 17

Using an online tool, we have;

x = y

From equation (1), we have;

2·y² - 3·y - 2 = 0

∴ y = 2, or y = -1/2

Where y = 2, we have;

x = y = 2

Therefore, the point b = (2, 2).