The diagonals of parallelogram ABCD intersect at E. If AE = x + 4 and AC- 5x - 10, find the value of AC.
AC = 2 AE
If AC- 5x - 10 means AC = 5 x - 10
then
AC = 2 AE
5 x - 10 = 2 ( x + 4 )
5 x - 10 = 2 x + 8
Add 10 to both sides
5 x = 2 x + 18
Subtract 2 x to both sides
3 x = 18
x = 18 / 3
x = 6
AC = 5 x - 10
AC = 5 ∙ 6 - 10 = 30 - 10 = 20
To find the value of AC, we need to utilize the given information about AE and AC.
First, notice that the diagonals of a parallelogram bisect each other. This means that each diagonal divides the other diagonal into two congruent segments. In other words, AE is equal to CE, and BE is equal to DE.
Let's consider the given information: AE = x + 4 and AC = 5x - 10.
Since AE is equal to CE, we have AE = CE = x + 4.
Now, we can set up an equation by adding the lengths of the two concurrent diagonals:
AE + CE = AC + AC.
Substituting the given values, we have:
x + 4 + x + 4 = 5x - 10 + 5x - 10.
Simplifying the equation:
2x + 8 = 10x - 20.
To isolate the variable x, we can rearrange the equation:
2x - 10x = -20 - 8.
Combine like terms:
-8x = -28.
Divide both sides by -8:
x = -28 / -8.
Simplifying further:
x = 7/2.
Now, we can substitute the value of x back into the equation to find AC:
AC = 5x - 10.
Substituting x = 7/2:
AC = 5*(7/2) - 10.
AC = 35/2 - 10.
To find the common denominator and simplify the equation, we multiply 2 by 5:
AC = 35/2 - 20/2.
AC = 15/2.
Therefore, the value of AC is 15/2.