Answer:
x = 3
y = 23
z = 4
Explanation:
To find the values of x, y, and z in the rhombus, we need to use the properties of a rhombus. One of the properties of a rhombus is that its opposite angles are equal. Another property is that its diagonals are perpendicular bisectors of each other.
Let's label the vertices of the rhombus as A, B, C, and D, and the midpoint of diagonal AC as M. We can use the given information to write some equations:
First, we know that diagonal AC is perpendicular to diagonal BD, so we have:
-2x + 6 = 0
Solving for x, we get:
x = 3
Next, we know that diagonal AC bisects angle ACD, so we have:
y = 46/2 = 23
Now, we can use the fact that AM is the perpendicular bisector of CD to find z. Since M is the midpoint of AC, we have:
AM = MC
Using the distance formula, we can find the lengths of AM and MC in terms of z:
AM = sqrt((2z-8)^2 + (5y-4)^2)
MC = sqrt((8-2z)^2 + (5y-4)^2)
Setting these two expressions equal to each other, we get:
sqrt((2z-8)^2 + (5y-4)^2) = sqrt((8-2z)^2 + (5y-4)^2)
Simplifying this equation, we get:
4z - 16 = 16 - 4z
8z = 32
z = 4
Therefore, the values of x, y, and z are:
x = 3
y = 23
z = 4
So, the solution is x=3, y=23, and z=4.