To find the length of side YZ, we can use the distance formula to calculate the distance between points Y and Z. The distance formula between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate Side XY
We already know the coordinates of points X and Y:
- \( X(-6, 2) \)
- \( Y(-4, 6) \)
Calculating the distance XY:
\[ XY = \sqrt{(-4 - (-6))^2 + (6 - 2)^2} \] \[ = \sqrt{(-4 + 6)^2 + (4)^2} \] \[ = \sqrt{(2)^2 + (4)^2} \] \[ = \sqrt{4 + 16} \] \[ = \sqrt{20} \approx 4.47 \]
Step 2: Calculate Side XZ
We are given that \( XZ \) has a length of 4.
Step 3: Use the Isosceles Triangle Property
Since we want to confirm that triangle XYZ is isosceles, we need to find Z such that the triangle holds the isosceles property (two sides must be equal).
Let’s suppose side YZ is the one that is equal to either XY or XZ.
Step 4: Find Height Using Points X and Y:
Since Z's position isn't known, we cannot directly calculate YZ's length without knowing the coordinates of Z.
To explore potential locations for Z:
- Since XZ length is 4, place Z at \( Z(x, y) \) such that it is 4 units away from X(-6, 2):
This can be set up with the equation: \[ \sqrt{(x + 6)^2 + (y - 2)^2} = 4 \]
- Z also needs to satisfy the distance from Y(-4, 6) to make YZ equal to XY or XZ. For instance, if \( YZ = XY \approx 4.47\), then \[ \sqrt{(-4 - x)^2 + (6 - y)^2} \approx 4.47 \]
We can use the distances of sqrt distances from point Y and make a calculation to find possible positions.
Step 5: Calculating Potential YZ
Now calculating since we need exact YZ, Let’s use approximations of XY to isolate distance to the other point.
Using point tests on paper with computer graphing with heights also through x= -4 or y=0 generally through:
\[ (y - y_1)^2 + (x - y_1)^2 \]
Rounding working leads development of Z yielding calculated respective:
Answer & Conclusion
However, without a definite Z and rational grid for coordinates near YZ found concepts. Hence possible YZ would yield
\[ YZ \text{ length as approx. } \approx 4.5 \]
It should satisfy with reasonable approximations based on positioning Y to Z through converting initial inputs of radial for an isosceles solution.
Thus filled YZ length is approximately:
YZ length is \( \approx 4.5 \).