For diagonals of square Abcd intersect at e . If ae =2x+6 and bd =6x-10 find ac
The diagonals of a square bisect each other into 4 equal lengths.
AE = 2x+6 is half of a diagonal, but
BD = 6x-10 is the entire diagonal, so its half would be 3x-5
then 3x-5 = 2x+6
x = 11
so AC = BD = 6x-10 = 6(11) - 10 = 56
It's 56
The diagonals of square ABCD intersect at E. If AE = 2x + 6 and BD = 6x – 10, find AC.
In 13-16, the diagonals of rectangle ABCD intersect at E in the diagram to the right.
13. Let and . Find AC, BD, AE, and BE.
Thank you so much!
Well, it seems we've stumbled upon a geometry problem! To find the length of AC, we can use the fact that diagonals of a square bisect each other.
Since AE and BD are diagonals of the square ABCD and they intersect at E, we can say that AE is equal to ED, and similarly, BE is equal to EC.
So, if AE is 2x + 6, then ED must also be 2x + 6 (since they're equal).
And if BD is 6x - 10, then BE must be (6x - 10)/2, which simplifies to 3x - 5.
Now, since BE is equal to EC, we can set up an equation:
BE = EC
3x - 5 = 2x + 6
By subtracting 2x from both sides, we get:
x - 5 = 6
By adding 5 to both sides, we get:
x = 11
Now that we've found x, we can substitute it back into AE to find the length of AC:
AE = 2x + 6
AE = 2(11) + 6
AE = 22 + 6
AE = 28
Therefore, the length of AC is 28.
To find the length of diagonal AC in the square ABCD, we can use the fact that diagonals of a square are perpendicular bisectors of each other. This means that diagonal AC divides diagonal BD into two equal halves.
Let's set up an equation to find the length of diagonal AC.
We are given that AE = 2x + 6 and BD = 6x - 10. Since diagonal AC divides diagonal BD into two equal halves, we can set up the following equation:
AE + EC = BD / 2
Substituting the given values, we have:
2x + 6 + EC = (6x - 10) / 2
Now, let's solve for EC:
2x + 6 + EC = 3x - 5
To isolate EC, we subtract 2x from both sides:
6 + EC = x - 5
Next, we add 5 to both sides:
11 + EC = x
Now we have an expression for EC in terms of x. However, we want to find the length of diagonal AC, not just the value of EC. Since diagonal AC divides the square into two congruent right triangles, we can use the Pythagorean theorem to relate the lengths of AC, EC, and AE:
AC^2 = AE^2 + EC^2
Substituting the given values, we have:
AC^2 = (2x + 6)^2 + (11 + EC)^2
Simplifying further, we have:
AC^2 = (2x + 6)^2 + (11 + x)^2
Expanding the square terms, we get:
AC^2 = (4x^2 + 24x + 36) + (x^2 + 22x + 121)
Combining like terms, we have:
AC^2 = 5x^2 + 46x + 157
Now, to find the length of diagonal AC, we need to take the square root of both sides:
AC = √(5x^2 + 46x + 157)
So, the length of diagonal AC in terms of x is √(5x^2 + 46x + 157).