91Ó°ÊÓ

The Pythagorean Theorem is a fundamental concept in geometry. It helps us find unknown side lengths in right triangles. The theorem is expressed as: \( c^2 = a^2 + b^2 \). Here, \( c \) is the hypotenuse, the longest side opposite the right angle. The legs, \( a \) and \( b \), form the right angle.

In an isosceles triangle, when an altitude is drawn, it divides the triangle into two right triangles. For the exercise at hand, the base of the triangle measures 32 cm. The altitude splits this base into two equal segments of 16 cm each. These segments act as one of the legs in our right triangle. The hypotenuse is one of the equal sides, measuring 42 cm.

Applying the Pythagorean Theorem:
\[42^2 = 16^2 + h^2\]
Solve this to find the height \( h \). This calculation gives us the height of the isosceles triangle, essential for further trigonometric applications.

Trigonometric functions come into play when we need to find angles in triangles. They're based on ratios between sides of right triangles. The primary functions are sine, cosine, and tangent.

In the case of this problem, you use the sine function. Sine helps us find angles when we know the length of the opposite side and the hypotenuse. It's defined as:

• \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

With the triangle split, the height becomes the opposite side, and the side length of 42 cm is the hypotenuse. You find the angle \( \theta \), which is one of the base angles, using the sine inverse function:
\[\theta = \arcsin\left(\frac{38.83}{42}\right) \approx 71.12^\circ\]
These base angles are equal due to the properties of an isosceles triangle.

Understanding triangle angles is crucial for solving such problems. Each triangle has a total angle sum of \(180^\circ\). In an isosceles triangle, there are two equal angles and a vertex angle.

Once we've calculated the base angles using trigonometric functions, finding the vertex angle becomes a matter of basic arithmetic. For isosceles triangles, use the formula:

• \( 180^\circ = 2\theta + \text{Vertex Angle} \)

Substitute the known values and simplify:
\[180^\circ = 2 \times 71.12^\circ + \text{Vertex Angle}\]
This allows us to easily find the vertex angle, \(37.76^\circ\).