Solve each problem. See Examples 1–4. Altitude of a Triangle Find the altitude of an isosceles triangle having base 184.2 cm if the angle opposite the base is 68°44'.

Verified step by step guidance

1. Convert the angle from degrees and minutes to decimal degrees. The angle is 68°44', which can be converted by adding 44/60 to 68.
2. Recognize that the altitude of the isosceles triangle forms two right triangles. The altitude is the opposite side to the angle at the base.
3. Use the trigonometric function tangent, which relates the opposite side (altitude) to the adjacent side (half of the base). The formula is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
4. Calculate the adjacent side, which is half of the base: \( \frac{184.2}{2} \).
5. Rearrange the tangent formula to solve for the opposite side (altitude): \( \text{opposite} = \tan(\theta) \times \text{adjacent} \). Substitute the values to find the altitude.

Key Concepts

Isosceles Triangle Properties

An isosceles triangle has two sides of equal length and two equal angles opposite those sides. The altitude from the vertex angle to the base bisects the base and creates two right triangles. Understanding these properties is essential for solving problems involving isosceles triangles, particularly when calculating heights or areas.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this problem, the sine function can be used to find the altitude by relating it to the angle opposite the base and the length of the base. Mastery of these ratios is crucial for solving various triangle-related problems.

Angle Conversion

In trigonometry, angles can be expressed in degrees, minutes, and seconds or in decimal degrees. The angle given in the problem, 68°44', needs to be converted into decimal degrees for easier calculation with trigonometric functions. Understanding how to convert between these formats is important for accurate computations.