91Ó°ÊÓ

A right triangle is a fundamental shape in geometry. It has one angle of 90 degrees and two other shorter angles that add up to 90 degrees.
Understanding right triangles makes it easier to solve many geometry problems, including those involving elevation angles.
In the given exercise, there are two right triangles, and they help us figure out the necessary distances and heights using trigonometry.
Here's how:

• In triangle APM, PM is the vertical height, and AM is the horizontal distance.
• In triangle BPM, PM remains the height, and BM is the new horizontal distance from point B to directly under the peak (M).

Breaking down the mountain problem into right triangles simplifies the process of solving it.

The angle of elevation is the angle formed by the line of sight when looking at an object above the horizontal level. In the mountain problem, this concept plays a key role.
At point A, the angle of elevation to the peak is 22 degrees. At point B, it is 38 degrees. These angles allow us to use trigonometric functions to find missing lengths in the triangles.
Using the tan function, which is the ratio of the opposite side (height) over the adjacent side (base), we calculate distances and heights efficiently.
Knowing these angles helps us set up our equations for solving the problem.

The Pythagorean theorem is used to relate the sides of a right triangle. The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is:
\[a^2 + b^2 = c^2\]
In our mountain exercise, once we have the height (h) and the horizontal distance from B (x - 220), we can use the Pythagorean theorem to find the distance from B to the peak (BP):
\[BP = \sqrt{( (x - 220)^2 + h^2 )}\]
This distance gives us the direct line Shannon and Erin need to hike to reach the peak from point B.

Trigonometric functions are essential for solving right triangle problems. In our exercise, we use the tangential function, defined as:
\[ \tan(\theta) = \frac{opposite}{adjacent} \]
For triangle APM:
\[ \tan(22^{\circ}) = \frac{h}{x} \]
Simplifies to:
\[ h = x \cdot \tan(22^{\circ}) \]
For triangle BPM:
\[ \tan(38^{\circ}) = \frac{h}{x - 220} \]
Simplifies to:
\[ h = (x - 220) \cdot \tan(38^{\circ}) \]
By equating these equations, we solve for the horizontal distance (x). This is a crucial step that ties our entire solution together and helps us find the peak height (h) and the total distance from B to the peak (BP).
Using trigonometric functions and setting up the right equations enable us to solve complex geometry problems systematically.