find lengths of diagonals AC and BD for the parallelogram where EB=x+17, ED=3y+2, EA=8x+1, and EC=6y-11
You know the diagonals bisect each other, so
x+17 = 3y+2
8x+1 = 6y-11
so solve for x and y, then you can get the lengths.
To find the lengths of diagonals AC and BD for the parallelogram, we will use the properties of parallelograms.
In a parallelogram, opposite sides are congruent, and opposite angles are congruent.
First, let's find the values of x and y from the given information.
From the properties of a parallelogram, we can conclude that EB = AD, so we have:
x + 17 = 8x + 1
Simplifying this equation, we get:
7x = 16
x = 16/7
Similarly, EC = AB, so we have:
6y - 11 = 8x + 1
Substituting the value of x (16/7) into this equation, we have:
6y - 11 = 8(16/7) + 1
Simplifying this equation, we get:
6y - 11 = 128/7 + 7/7
6y - 11 = 135/7
6y = 135/7 + 77/7
6y = 212/7
y = 212/42
y = 106/21
Now, we have found the values of x and y.
To find the length of diagonal AC, we can use the distance formula. The coordinates of the points A and C are (x, y) in the form (8x + 1, 6y - 11).
Using the distance formula, we have:
AC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
AC = sqrt((8x + 1 - x)^2 + (6y - 11 - y)^2)
AC = sqrt((7x + 1)^2 + (5y - 11)^2)
AC = sqrt((7(16/7) + 1)^2 + (5(106/21) - 11)^2)
AC = sqrt((17)^2 + (3)^2)
AC = sqrt(289 + 9)
AC = sqrt(298)
So, the length of diagonal AC is sqrt(298).
To find the length of diagonal BD, we can use the same approach. The coordinates of the points B and D are (x, y) in the form (x + 17, 3y + 2).
Using the distance formula, we have:
BD = sqrt((x2 - x1)^2 + (y2 - y1)^2)
BD = sqrt((x + 17 - x)^2 + (3y + 2 - y)^2)
BD = sqrt((17)^2 + (2y + 2)^2)
BD = sqrt(289 + (2(106/21) + 2)^2)
BD = sqrt(298)
So, the length of diagonal BD is also sqrt(298).
Therefore, the lengths of diagonals AC and BD for the given parallelogram are both sqrt(298).