ABCD is a rectangle with B(-4,2) andD (10, 6). find the coordinates of A. Please help and tell me how you got it so I can understand
So easiest way to understand it might be to make a sketch.
Looks like BD is going to be the diagonal of the rectangle
So you should have no problem seeing that
A is (-4,6)
C is (10, 2)
I'm sorry I put it wrong B(-4,6) c(-4,2) d(10,2)
Again, sketch it and do it the way I suggested by just looking at the points.
notice C and D have the same y, so CD is a horizontal line
and B and C have the same X, so BC must be a vertical line
It is even easier than the way you first had it.
Okay thank you do much!:)
To find the coordinates of point A, we need to understand some properties of a rectangle. In a rectangle, opposite sides are parallel and equal in length. Additionally, the diagonals bisect each other.
Given the coordinates of points B and D, we can determine the length of side AD and the slope of line AD. Then, using these values, we can calculate the coordinates of point A.
Step 1: Find the length of side AD
The length of side AD can be calculated using the distance formula:
D = √((x2 - x1)^2 + (y2 - y1)^2)
Let's label the coordinates of point A as (x, y). Using the coordinates of points B and D:
AD = √((10 - x)^2 + (6 - y)^2)
Step 2: Find the slope of line AD
The slope of a line passing through two points, (x1, y1) and (x2, y2), can be calculated using the formula:
m = (y2 - y1)/(x2 - x1)
In this case, the slope of AD can be calculated using points B and D:
m = (6 - 2)/(10 - (-4))
Step 3: Calculate the coordinates of point A
Now, we have two equations:
AD = √((10 - x)^2 + (6 - y)^2) -> equation (1)
m = (6 - 2)/(10 - (-4)) -> equation (2)
For a rectangle, the diagonals bisect each other. So, the midpoint of BD will be the midpoint of AC. Since we know the coordinates of B and D, we can easily calculate the midpoint of BD as:
Midpoint(x_mid, y_mid) = ((x1 + x2)/2, (y1 + y2)/2)
Let's substitute the coordinates of B and D:
Midpoint(x_mid, y_mid) = ((-4 + 10)/2, (2 + 6)/2)
Simplifying this:
Midpoint(x_mid, y_mid) = (6/2, 8/2) = (3, 4)
Now we have the coordinates of point C as well. The midpoint of AC is also (3, 4), as the diagonals of a rectangle bisect each other.
Using equation (2), we know:
m = (6 - 2)/(10 - (-4)) = (4)/(14) = 2/7
Since the slope of AD is perpendicular to the slope of AC, their product is -1. So we have:
m * m_AD = -1
(2/7) * m_AD = -1
Simplifying:
m_AD = -7/2
Now, we can use the point-slope form of a line to find the equation of line AD:
(y - y1) = m(x - x1)
(y - 4) = (-7/2)(x - 3)
Expanding this equation:
(y - 4) = (-7/2)x + (21/2)
y = (-7/2)x + (21/2) + 4
y = (-7/2)x + (21/2) + (8/2)
y = (-7/2)x + (29/2)
Now we can use equation (1) to find the coordinates of point A by substituting y in terms of x in the equation of line AD:
AD = √((10 - x)^2 + ((-7/2)x + (29/2) - y)^2)
Simplifying this equation, we have:
AD = √((10 - x)^2 + ((-7/2)x + (29/2) - ((-7/2)x + (21/2)))^2)
Simplifying further:
AD = √((10 - x)^2 + ((-7/2)x + (29/2) - (-7/2)x - (21/2))^2)
AD = √((10 - x)^2 + (8/2)^2)
AD = √((10 - x)^2 + 16)
AD = √(100 - 20x + x^2 + 16)
AD = √(x^2 - 20x + 116)
Since AD is equal to AC, we can equate the equations:
x^2 - 20x + 116 = (10 - x)^2 + 16
Expanding both sides, we have:
x^2 - 20x + 116 = 100 - 20x + x^2 + 16
Canceling like terms, we have:
-20x + 116 = -20x + 116
This equation tells us that regardless of the value of x, the left side will always be equal to the right side. In other words, we have an identity. Therefore, the value of x doesn't matter in this case.
To find the value of y, we can substitute x = 3 (the x-coordinate of the midpoint of AC) into the equation of line AD:
y = (-7/2)(3) + (29/2)
y = (-21/2) + (29/2)
y = 8/2
y = 4
Therefore, the coordinates of point A are (3, 4).
In conclusion, the coordinates of point A are (3, 4) for the given rectangle ABCD with B(-4, 2) and D(10, 6).