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. Identify the given elements: the base of the isosceles triangle is 184.2 cm, and the angle opposite the base is 68°44'. Since the triangle is isosceles, the two equal sides meet at this angle.
2. Convert the angle 68°44' into decimal degrees for easier calculation: 68° + (44/60)°.
3. Draw the altitude from the vertex opposite the base to the base, which will bisect the base into two equal segments of length \(\frac{184.2}{2}\) cm each.
4. Use the right triangle formed by the altitude, half the base, and one of the equal sides. Apply the trigonometric function sine to relate the altitude (opposite side) to the hypotenuse (equal side). The altitude \(h\) can be found using \(h = \left( \frac{184.2}{2} \right) \times \tan\left( \frac{68.7333}{2} \right)\), where \(68.7333\) is the decimal form of the angle.
5. Calculate the altitude \(h\) by evaluating the tangent of half the angle and multiplying by half the base length.

Key Concepts

Properties of Isosceles Triangles

An isosceles triangle has two equal sides and two equal angles opposite those sides. The altitude from the vertex angle bisects the base and creates two congruent right triangles, which helps in calculating unknown lengths using trigonometric ratios.

Trigonometric Ratios in Right Triangles

Trigonometric ratios such as sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. These ratios allow calculation of unknown side lengths or angles when one side and one angle are known.

Angle Conversion and Use of Degrees and Minutes

Angles given in degrees and minutes (e.g., 68°44') must be converted to decimal degrees or used directly in trigonometric calculations. Understanding this notation is essential for accurate computation of trigonometric functions.