if AB=CD, AB is 12-y, CD = 3x-4y, BD = 18, AC = x+y, what is x, y, and AC
No way to tell, unless ABCD is a parallelogram. In that case, AB=CD and AC=BD
12-y = 3x-4y
x+y = 18
(x,y) = (11,7)
To find the values of x, y, and AC, we can set up a system of equations using the given information.
1) AB = CD:
We are given that AB is 12-y and CD is 3x-4y. This means 12-y = 3x-4y.
2) BD = 18:
We are given that BD equals 18, which represents the length of the line segment BD.
3) AC = x+y:
We are given that AC is equal to x+y, where x and y are variables we want to determine.
Now, let's solve this system of equations step by step:
First, let's simplify the equation AB = CD:
12-y = 3x-4y
To simplify further, let's get all the y terms on one side of the equation:
12-y+4y = 3x
12+3y = 3x
Next, let's solve for x in terms of y:
3x = 12+3y
x = (12+3y)/3
x = 4+y
Now, let's consider the equation AC = x+y:
AC = x+y
AC = (4+y)+y
AC = 4+2y
We are given that BD = 18, so we can set up another equation using the Pythagorean theorem since we have a right triangle:
BD^2 = AB^2 + AD^2
Substituting the values we know, we get:
18^2 = (12-y)^2 + (x+y)^2
Squaring the terms:
324 = (12-y)(12-y) + (x+y)(x+y)
324 = 144 - 24y + y^2 + x^2 + 2xy + y^2
Simplifying:
0 = 288 - 24y + y^2 + x^2 + 2xy
Now, we have a system of two equations:
1) 12+3y = 3x
2) 324 = 288 - 24y + y^2 + x^2 + 2xy
To solve this system, we can substitute the expression for x from equation 1) into equation 2):
324 = 288 - 24y + y^2 + (4+y)^2 + 2(4+y)y
Simplifying further, we get:
324 = 288 - 24y + y^2 + (16+8y+y^2) + 2y(4+y)
Combining like terms:
324 = 288 - 24y + y^2 + 16 + 8y + y^2 + 8y + 2y^2
Simplifying again:
0 = 2y^2 + 10y + (-20)
Now, we have a quadratic equation:
2y^2 + 10y - 20 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Solving the quadratic equation will give us the values of y. Once we have the value(s) of y, we can substitute them back into equation 1) to find x, and then substitute both x and y values back into equation 3) to find AC.