Problem 56 - Algebra and Trigonometry

Question

An isosceles triangle has an area of \(24 \mathrm{cm}^{2},\) and the angle between the two equal sides is 5\(\pi / 6 .\) What is the length of the two equal sides?

Step-by-Step Solution

Answer

The length of the two equal sides is approximately \(12 \text{ cm}\).

Step 1: Understand the Triangle

We have an isosceles triangle with an area of \(24 \text{ cm}^2\) and one angle between the equal sides is \(\frac{5\pi}{6}\). We need to find the length of the equal sides.

Step 2: Use Area Formula for Triangle

The area \(A\) of a triangle with two sides \(a\) and \(b\) forming an angle \(\theta\) is given by \(A = \frac{1}{2}ab\sin(\theta)\). In this case, both sides are equal, so they are \(a = b\), and \(\theta = \frac{5\pi}{6}\).

Key Concepts

Triangle Area Formula

Understanding the formula for finding the area of a triangle is essential, especially in solving problems involving isosceles triangles. The general area formula for any triangle involves its base and height. However, when two sides and the included angle are known, we use a different approach.

The formula is:

• For any triangle with sides of length \(a\) and \(b\), and an angle \(\theta\) between them, the area \(A\) can be calculated using:
• \(A = \frac{1}{2}ab\sin(\theta)\)

For our problem, since it's an isosceles triangle, the two sides are equal. This simplifies our formula to \(A = \frac{1}{2}a^2\sin(\theta)\) where \(a\) is the length of the equal sides. By substituting the known values, you can easily calculate the length of the sides. This makes it convenient to solve the area when the height is not directly known.

Trigonometry

Trigonometry becomes very handy when dealing with triangles, especially for solving problems where the direct measurements of heights are not available. It's all about relationships between the angles and lengths of the triangles.

In the context of our problem, we use the sine function, which is part of the basic trigonometric functions. The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Here’s why it's important:

• When you know two sides of a triangle and the angle between them, trigonometry helps find the area.
• The sine function allows the expression of height in terms of known and equal sides \(a\) and the angle \(\theta\) in our formula \(\sin(\theta)\).

This connection is crucial as it allows the computation of areas beyond basic geometric expectation, hence providing more versatility in triangle calculations.

Angle Between Sides

In any triangle where we aim to find the area using known side lengths and an angle, the angle between the sides is paramount. The success of calculating the area via our formula depends on this angle.

For isosceles triangles, like in our example, understanding the angle between the equal sides is key:

• This angle is what determines how 'wide' the triangle is at the base, affecting its overall shape and hence the effective area.
• In our problem, the angle is given as \(\frac{5\pi}{6}\), which can be converted to degrees for better understanding: approximately 150 degrees.

This larger angle signals a more stretched-out triangle, as opposed to a more acute, steeper triangle. Therefore, understanding such characteristics helps in visualizing the context of the problem and efficiently solving for the unknown side lengths.