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 |XZ|

Given XYZ, WTF is PQR?

I mean |<XYZ|

XZ^2 = 6^2 + 8^2 - 2*6*8cosY

cosY is either sqrt(3)/2 or -sqrt(3)/2

XZ^2 = 100-96*.866
XZ = 4.1
or
XZ^2 = 100+96*.866
XZ = 13.5

To find the possible lengths of |XZ| in triangle XYZ, we need to use the triangle area formula and trigonometry.

Let's start by calculating the height of the triangle. The area of a triangle can be calculated using the formula: Area = (1/2) * base * height.

In this case, the base is |XY|, which is 8 cm, and the area is 12 cm^2. Let's substitute these values into the formula and solve for the height:

12 cm^2 = (1/2) * 8 cm * height

Multiplying both sides by 2 gives us:

24 cm^2 = 8 cm * height

Dividing both sides by 8 cm:

height = 3 cm

Now that we have the height, we can use trigonometry to find the lengths of |XZ|. Since we are given the angles |<PQR|= 30 degrees and 150 degrees, we know that angle |<YXZ| is the supplementary angle to |<PQR|.

To find the lengths of |XZ|, we can use the sine function:

sin(angle) = opposite / hypotenuse

Since |XZ| is the hypotenuse and the height we calculated earlier is the opposite side, the equation becomes:

sin(angle) = 3 cm / |XZ|

We can solve this equation to find the possible lengths of |XZ|:

1. For angle |<PQR| = 30 degrees:

sin(30 degrees) = 3 cm / |XZ|

Using a calculator, we find that sin(30 degrees) = 0.5. Therefore:

0.5 = 3 cm / |XZ|

Multiply both sides by |XZ|:

0.5 * |XZ| = 3 cm

Divide both sides by 0.5:

|XZ| = 6 cm

So, one possible length for |XZ| is 6 cm when angle |<PQR| = 30 degrees.

2. For angle |<PQR| = 150 degrees:

sin(150 degrees) = 3 cm / |XZ|

Using a calculator, we find that sin(150 degrees) = -0.5. Since sine is negative in the second and third quadrants, we need to consider the absolute value of the sin value:

0.5 = 3 cm / |XZ|

Proceeding with the same steps as before, we have:

0.5 * |XZ| = 3 cm

|XZ| = 6 cm

Again, |XZ| = 6 cm is a possible length when angle |<PQR| = 150 degrees.

Therefore, the two possible lengths for |XZ| in triangle XYZ are 6 cm when angle |<PQR| is either 30 degrees or 150 degrees.