If rectangle ABCD has vertices A(6, 2), B(2, 2), and C(0, 6), explain how to find the coordinate for the fourth vertex. Then describe how to verify that the angles of ABCD are right angles.
Can somebody just post all the answers???? I really need help this my last assignment and if I get a f I'm gonna fail...
mmmhhh, did you plot your points?
Can you see any way they could form a rectangle?
can someone answers this please
To find the coordinate of the fourth vertex of rectangle ABCD, we can use the properties of rectangles. Rectangles are quadrilaterals with four right angles, meaning that opposite sides are parallel and equal in length.
Since AB and CD are parallel and equal in length, we can find the equation of the line passing through points A and B. Then, using the slope-intercept form of a line, we can determine the equation for the line segment AB.
1. Find the slope (m) of AB:
- m = (y2 - y1) / (x2 - x1)
- m = (2 - 2) / (2 - 6)
- m = 0
2. Use the point-slope form of a line (y - y1) = m(x - x1) with one of the points on AB (e.g., A):
- (y - 2) = 0(x - 6)
- y - 2 = 0
- y = 2
So, the equation of line AB is y = 2.
Since the opposite sides of a rectangle are parallel, the equation of line CD will also be y = 2.
Now, to find the x-coordinate of the fourth vertex, we need to find where the line y = 2 intersects the x-axis. Since y = 2 represents a horizontal line, it intersects the x-axis at y = 0.
Substituting y = 0 into the equation y = 2, we get:
0 = 2
This implies that the line y = 2 does not intersect the x-axis.
Therefore, there is no y-intercept for the line CD, which means it is a vertical line. The x-coordinate of the fourth vertex is given by the x-coordinate of point C, which is 0.
Hence, the coordinate of the fourth vertex (D) is (0, y), where y can be any value.
To verify that the angles of ABCD are right angles, we can compute the slopes of the lines AB, BC, CD, and DA.
1. Slope of AB:
- Already calculated as 0
2. Slope of BC:
- m = (y2 - y1) / (x2 - x1)
- m = (2 - 2) / (0 - 2)
- m = 0
3. Slope of CD:
- Vertical lines have undefined slopes, but we know it is a vertical line.
4. Slope of DA:
- m = (y2 - y1) / (x2 - x1)
- m = (2 - 6) / (0 - 6)
- m = -4 / -6
- m = 2/3
Since the slopes of AB and CD are zero, and the slopes of BC and DA are negative reciprocals of each other (-2/3 and 2/3), we can conclude that the opposite sides of ABCD are perpendicular to each other, making each angle a right angle.