To solve for the lengths $x$ and $y$ in the given triangle diagram, we can break the problem down step by step.

Information from the diagram:

• The total length of the base of the truss is $11.2 \, \text{ft}$.
• One section of the base is $4.8 \, \text{ft}$, corresponding to the leg next to the angle $42^\circ$.
• The hypotenuse of this right triangle is $6.3 \, \text{ft}$ (labeled in the diagram).
• We have two unknowns: $x$ (the vertical height of the triangle on the right) and $y$ (the diagonal hypotenuse on the left triangle).

To find $x$ (the height):

We can use trigonometry in the right triangle on the right-hand side. Specifically, we can apply the sine and cosine functions:

• Using the angle $42^\circ$ and the known side $6.3 \, \text{ft}$ (the hypotenuse), we can use the sine function to find $x$, since $x$ is opposite the angle: $\sin(42^\circ) = \frac{x}{6.3}$ Solving for $x$: $x = 6.3 \times \sin(42^\circ)$

$x \approx 6.3 \times 0.6691 \approx 4.21 \, \text{ft}$

To find $y$ (the diagonal on the left side):

Next, we need to find $y$. Since the entire base of the triangle is $11.2 \, \text{ft}$, and we already know the right portion of the base is $4.8 \, \text{ft}$, the left portion of the base will be: $\text{Left base} = 11.2 - 4.8 = 6.4 \, \text{ft}$ Now we have a right triangle on the left side where we know the base is $6.4 \, \text{ft}$ and the height is $4.21 \, \text{ft}$ (the same as $x$). To find $y$ (the hypotenuse of the left triangle), we can use the Pythagorean theorem: $y^2 = 6.4^2 + 4.21^2$

$y^2 = 40.96 + 17.72$ $y^2 = 58.68$ $y \approx \sqrt{58.68} \approx 7.66 \, \text{ft}$

Final Answer:

• $x \approx 4.21 \, \text{ft}$
• $y \approx 7.66 \, \text{ft}$

Let me know if you would like further details or have any questions.

Here are some related questions for further practice:

1. How would the result change if the angle was different, say $30^\circ$?
2. What is the area of the entire truss?
3. How could we find the perimeter of the truss?
4. If the height doubled, how would that impact the lengths $x$ and $y$?
5. What would the length of $y$ be if the base on the left side was longer?

Tip: When solving right triangles, it's helpful to remember the basic trigonometric ratios—sine, cosine, and tangent—depending on which sides and angles you know.