In triangle XYZ, |XY|= 8 cm and |YZ|=6m.

The area of triangle XYZ is 12cm^2. The |<PQR|= 30 degrees, 150 degrees.
Find the two possible values of |XZ, correct 1 decimal place

To find the possible values of |XZ (length of side XZ) in triangle XYZ, we can use the Law of Cosines. The formula for the Law of Cosines is:

c^2 = a^2 + b^2 - 2ab * cos(C)

Where:
- c is the length of the side opposite angle C
- a and b are the lengths of the other two sides of the triangle
- C is the angle between the sides a and b

In this case, the angle between sides XY and YZ is not given, but we know the area of the triangle. We can use the area formula of a triangle to find this angle. The formula for the area of a triangle is:

Area = 0.5 * base * height

We can rearrange this formula to solve for the height:

height = (2 * Area) / base

Given that the area of triangle XYZ is 12 cm², and the base is the length of side XY (8 cm), we can solve for the height:

height = (2 * 12 cm²) / 8 cm
height = 3 cm

So, the height of triangle XYZ is 3 cm. Now we can use this height to split triangle XYZ into two right triangles, XYH and HYZ, where H is the foot of the height perpendicular to side XZ.

Now we have two right triangles, XYH and HYZ, where |XY| = 8 cm, |YZ| = 6 cm, and |HY| = 3 cm. To find the possible values of |XZ|, we will find the length of side XH and side HZ separately using the Pythagorean theorem.

For triangle XYH:
|XH|^2 = |XY|^2 - |HY|^2
|XH|^2 = 8 cm^2 - 3 cm^2
|XH|^2 = 64 cm^2 - 9 cm^2
|XH|^2 = 55 cm^2
|XH| = √55 cm (approximately 7.42 cm)

For triangle HYZ:
|ZH|^2 = |YZ|^2 - |HY|^2
|ZH|^2 = 6 cm^2 - 3 cm^2
|ZH|^2 = 36 cm^2 - 9 cm^2
|ZH|^2 = 27 cm^2
|ZH| = √27 cm (approximately 5.2 cm)

Therefore, the two possible values of |XZ| are approximately 7.4 cm and 5.2 cm (rounded to one decimal place).