To solve for the unknown length $x$ in the provided right triangle diagram, we need to apply principles from geometry and trigonometry. Let's analyze the given figure:

Step 1: Understand the setup

1. The triangle includes two right angles, meaning this is a composite of two smaller right triangles.

2. The larger triangle has:

   • Hypotenuse = $x$,
   • One leg = $4$ (vertical leg),
   • The base is the entire bottom side.

3. The smaller triangle inside has:

   • Hypotenuse = $8$,
   • Vertical leg = $4$,
   • A shared base (common with the larger triangle).

Step 2: Solve for the base of the smaller triangle

Using the Pythagorean theorem for the smaller triangle: $\text{(hypotenuse)}^2 = \text{(leg 1)}^2 + \text{(leg 2)}^2$ $8^2 = 4^2 + \text{(base)}^2$ $64 = 16 + \text{(base)}^2$ $\text{(base)}^2 = 48 \implies \text{base} = \sqrt{48} = 4\sqrt{3}$

Thus, the base of the smaller triangle is $4\sqrt{3}$.

Step 3: Solve for the base of the larger triangle

The larger triangle's base is the sum of the base of the smaller triangle and the segment to its right: $\text{Larger triangle base} = 4\sqrt{3} + 4$

Step 4: Solve for $x$ in the larger triangle

Using the Pythagorean theorem for the larger triangle: $x^2 = \text{(vertical leg)}^2 + \text{(base)}^2$ $x^2 = 4^2 + (4\sqrt{3} + 4)^2$

Expand $(4\sqrt{3} + 4)^2$: $(4\sqrt{3} + 4)^2 = (4\sqrt{3})^2 + 2(4\sqrt{3})(4) + 4^2$ $= 48 + 32\sqrt{3} + 16 = 64 + 32\sqrt{3}$

Now substitute this back: $x^2 = 16 + (64 + 32\sqrt{3})$ $x^2 = 80 + 32\sqrt{3}$

Step 5: Simplify

The length of $x$ is: $x = \sqrt{80 + 32\sqrt{3}}$

This is the simplified exact value. If needed, we can approximate it numerically: $x \approx 10.86 \, (\text{rounded to two decimal places}).$

Would you like further clarification or a deeper breakdown of any step?

Related Questions:

1. How can the Pythagorean theorem be used in different triangle configurations?
2. What are some tips for simplifying square roots with radicals like $\sqrt{48}$?
3. How do we know when to apply the Pythagorean theorem versus trigonometric ratios?
4. Can this problem be solved using trigonometric functions (e.g., sine, cosine)?
5. What is the geometric significance of the smaller triangle within the larger triangle?

Tip:

Always double-check your work by substituting the calculated values back into the original equations to ensure they hold true.