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The given diagram is a triangle with a horizontal base \(z\), and it includes angles of \(30^\circ\) and \(60^\circ\) at the lower corners. A perpendicular divides the triangle into two right triangles, with one side labeled as 100 feet and the others labeled \(x\), \(y\), and \(z\). We are tasked with finding the lengths of \(x\), \(y\), and \(z\).

### Let's break it down:

1. The large triangle is split into two right triangles.
   - One triangle has an angle of \(30^\circ\), and the other has an angle of \(60^\circ\).
   
2. The side opposite the right angle (the height) is given as 100 feet. We will use trigonometric ratios to find the lengths.

---

### Using Trigonometric Ratios:

#### Right Triangle with \(30^\circ\):
For the triangle with the \(30^\circ\) angle:
- \[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{100}{y}
\]
  \[
y = \frac{100}{\sin(30^\circ)} = \frac{100}{\frac{1}{2}} = 200 \text{ feet}
  \]
  
#### Right Triangle with \(60^\circ\):
For the triangle with the \(60^\circ\) angle:
- \[
\sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{100}{x}
\]
  \[
x = \frac{100}{\sin(60^\circ)} = \frac{100}{\frac{\sqrt{3}}{2}} = \frac{100 \times 2}{\sqrt{3}} = \frac{200}{\sqrt{3}} \approx 115.47 \text{ feet}
  \]

#### Finding \(z\) (the total base):
Since \(z\) is the total base of the triangle, it is composed of the horizontal legs of both right triangles. 
- For the triangle with \(30^\circ\):
  \[
  \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\text{adjacent}}{200}
  \]
  \[
  \text{adjacent (part of } z \text{)} = 200 \times \cos(30^\circ) = 200 \times \frac{\sqrt{3}}{2} = 100\sqrt{3} \approx 173.21 \text{ feet}
  \]
  
- For the triangle with \(60^\circ\):
  \[
  \cos(60^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\text{adjacent}}{x}
  \]
  \[
  \text{adjacent (other part of } z \text{)} = x \times \cos(60^\circ) = \frac{200}{\sqrt{3}} \times \frac{1}{2} = \frac{100}{\sqrt{3}} \approx 57.74 \text{ feet}
  \]

Now, the total base \(z\) is the sum of both parts:
\[
z = 100\sqrt{3} + \frac{100}{\sqrt{3}} \approx 173.21 + 57.74 = 230.95 \text{ feet}
\]

### Summary:
- \(x \approx 115.47\) feet
- \(y = 200\) feet
- \(z \approx 230.95\) feet

Would you like further details or have any questions about these steps?

Here are 5 relative questions:
1. How do we calculate the sine and cosine values for specific angles?
2. Can trigonometry be used in non-right triangles?
3. What is the significance of the angle sum in a triangle?
4. How would this approach change if the angles were different?
5. How can we verify the accuracy of these measurements using another method?

**Tip:** Always make sure your calculator is in the correct mode (degrees or radians) when solving trigonometric problems!

Find the exact lengths x, y, and z in the triangle given with angles 30°, 60°, and a side length of 100 feet.

Learn how to solve for the exact lengths x, y, and z in a triangle with given angles of 30°, 60°, and a side of 100 feet. Using trigonometric ratios (sine and cosine), this problem highlights essential geometry and trigonometry concepts. Suitable for students in Grades 10-11.

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