We are given a right triangle $ABC$ with $C = 90^\circ$, and the legs:
$a = 38$ ft, $b = 86$ ft.
We need to find:

• Hypotenuse $c$
• Angle $A$
• Angle $B$

Step 1: Find Hypotenuse $c$ using the Pythagorean Theorem

$c = \sqrt{a^2 + b^2}$ $c = \sqrt{(38)^2 + (86)^2}$ $c = \sqrt{1444 + 7396}$ $c = \sqrt{8840} \approx 94 \text{ ft}$

Step 2: Find Angle $A$ using Trigonometry

Using the tangent function: $\tan A = \frac{a}{b} = \frac{38}{86}$ $A = \tan^{-1} \left( \frac{38}{86} \right)$ $A \approx \tan^{-1} (0.4419)$ $A \approx 24^\circ$

Step 3: Find Angle $B$

Since the sum of angles in a triangle is $180^\circ$: $A + B + 90^\circ = 180^\circ$ $B = 180^\circ - 90^\circ - A$ $B = 180^\circ - 90^\circ - 24^\circ$ $B \approx 66^\circ$

Final Answers:

• $c \approx 94$ ft
• $A \approx 24^\circ$
• $B \approx 66^\circ$

Would you like a detailed breakdown of any step?

Related Problems:

1. Solve a right triangle where $a = 45$ ft and $b = 60$ ft.
2. Find the hypotenuse when $a = 12$ ft and $b = 35$ ft.
3. Given $c = 100$ ft and $a = 80$ ft, find $b$.
4. Find angles $A$ and $B$ when $a = 50$ ft and $c = 100$ ft.
5. Solve for all sides and angles when $A = 40^\circ$ and $b = 50$ ft.

Math Tip:

When solving right triangles, always check if trigonometry or the Pythagorean Theorem is the fastest method!