Final answer:
To find the length of ZY, we use the sine function in the right triangle XYW and the given lengths and angles to calculate ZY ≈ 1.60.
Step-by-step explanation:
The question asks to find the length of ZY in triangle XYZ where we know that ∠Y = 90°, ∠X = 4°, and an external angle ∠zwy = 74°.
Since we're given xw = 7.7 and the angles, we can find ∠Z using the fact that the sum of angles in a triangle is 180°. Thus, ∠Z = 180° - (∠Y + ∠X) = 180° - (90° + 4°) = 86°.
Knowing ∠Z and the external angle ∠zwy = 74°, we can now establish that ∠ZWY = ∠Z - ∠zwy = 86° - 74° = 12°. From the right triangle XYW, we can use the sine function to find ZY.
Sin(∠ZWY) = opposite/hypotenuse = ZY/XW => ZY = XW * Sin(∠ZWY).
Substituting the values gives ZY = 7.7 * Sin(12°).
Using a calculator, Sin(12°) approximates to 0.2079, so ZY ≈ 7.7 * 0.2079.
After calculation, ZY ≈ 1.60, to the nearest hundredth.