To solve for the exact lengths $x$, $y$, and $z$ in the given triangle, we'll use trigonometric relationships and the Pythagorean theorem. Here's the plan:

1. Identify the triangle components:

   • The triangle is split into two right triangles by the perpendicular from the vertex to the base.
   • The base $z$ is divided into two segments, one of which is 60 feet, and the other we can call $a$ (which is part of $z$).

2. Use trigonometric functions:

   • For the left right triangle:

     • The angle is $30^\circ$.
     • The height $100$ feet is the opposite side to the angle.
     • The base $a$ (part of $z$) is the adjacent side.

   • For the right right triangle:

     • The angle is $60^\circ$.
     • The height $100$ feet is again the opposite side to the angle.
     • The base $y - 60$ (other part of $z$) is the adjacent side.

3. Solve for the unknowns:

   • Using the tangent function, which is $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$: $\tan(30^\circ) = \frac{100}{a}$ $\tan(60^\circ) = \frac{100}{y - 60}$

   • Then, solve for $x$ using the Pythagorean theorem in one of the right triangles.

Let’s solve it step by step.

Step 1: Calculate $a$

$\tan(30^\circ) = \frac{100}{a}$ $a = \frac{100}{\tan(30^\circ)} = 100 \times \sqrt{3} \approx 173.21 \text{ feet}$

Step 2: Calculate $y - 60$

$\tan(60^\circ) = \frac{100}{y - 60}$ $y - 60 = \frac{100}{\tan(60^\circ)} = \frac{100}{\sqrt{3}} \approx 57.74 \text{ feet}$ $y = 60 + 57.74 \approx 117.74 \text{ feet}$

Step 3: Calculate $z$

$z = a + 60 = 173.21 + 60 = 233.21 \text{ feet}$

Step 4: Calculate $x$ using the Pythagorean theorem:

$x^2 = 100^2 + a^2$ $x^2 = 100^2 + (173.21)^2$ $x = \sqrt{10000 + 30000} \approx 200 \text{ feet}$

Final Answers:

• $x = 200$ feet
• $y = 117.74$ feet
• $z = 233.21$ feet

Would you like any additional details or have any questions about these calculations?

Here are five related questions to explore further:

1. How does the tangent function relate to the angles in different triangles?
2. Can the values of $x$, $y$, and $z$ be found using different trigonometric identities?
3. What is the significance of the Pythagorean theorem in right triangles?
4. How would the problem change if the height was different?
5. Can you derive the sine or cosine of angles using this setup?

Tip: When solving trigonometric problems, always ensure your calculator is in the correct mode (degrees or radians) corresponding to the angle measurements provided.