Problem 43 - Algebra 2

Question

An altitude inside a triangle forms angles of \(36^{\circ}\) and \(42^{\circ}\) with two of the sides. The altitude is 5 \(\mathrm{m}\) long. Find the area of the triangle.

Step-by-Step Solution

Answer

The area of the triangle can be calculated using the formula \( \frac{1}{2} * base * height \), where the base can be found by adding the lengths of the two sides of the smaller triangles: \( a + b = \frac{5 m}{\sin(36^{\circ})} + \frac{5 m}{\sin(42^{\circ})} \) and height is the given altitude.

Step 1: Identify Traingle Segments Using Trigonometric Principle

The triangle consists of two smaller right triangles formed by the altitude. We can calculate the lengths of the sides of these two triangles, which also form the base of the larger triangle, using the sine function. For the first right triangle, the altitude forms an angle of \(36^{\circ}\) with one side. Let's call the side adjacent to the \(36^{\circ}\) angle as \(a\), and the side opposite as \(h\), which is the altitude of 5 m. Therefore, \( \sin(36^{\circ}) = \frac{h}{a}\). From this we can calculate \(a\) as \(a = \frac{h}{\sin(36^{\circ})} = \frac{5 m}{\sin(36^{\circ})}\). Similarily, for the second right triangle, the altitude forms a \(42^{\circ}\) angle with one side. Let's call the side adjacent to the \(42^{\circ}\) angle as \(b\), the side opposite will be the same altitude of 5 m and so \(b = \frac{h}{\sin(42^{\circ})} = \frac{5 m}{\sin(42^{\circ})}\). Thus, we have calculated lengths of sides \(a\) and \(b\) of the two right triangles.

Step 2: Calculate the length of Larger Triangle's base

The base of the larger triangle is the sum of lengths of the sides \(a\) and \(b\). Thus, \(base = a + b = \frac{5 m}{\sin(36^{\circ})} + \frac{5 m}{\sin(42^{\circ})}\).

Step 3: Calculate the Area of the Triangle

The area of a triangle is calculated as half the product of its base and height. Here, the base of the larger triangle is the sum of \(a\) and \(b\), and its height is the altitude of 5 m. Therefore, the area of the triangle is \(Area = \frac{1}{2} * base * height = \frac{1}{2} * (\frac{5 m}{\sin(36^{\circ})} + \frac{5 m}{\sin(42^{\circ})}) * 5 m\).

Key Concepts

Sine Function

The sine function is a fundamental component in trigonometry. It is particularly useful when dealing with right-angled triangles. Let's break it down in simple terms. Imagine a right triangle where one of the angles is labeled as \( \theta \). The sine of angle \( \theta \), written as \( \sin(\theta) \), is the ratio between the length of the side opposite to \( \theta \) and the hypotenuse of the triangle.

• Opposite side - the side directly opposite to the angle in question.
• Hypotenuse - the longest side of the triangle, opposite the right angle.

This means if we know the angle \( \theta \) and either the opposite side or the hypotenuse, we can calculate the other side. This calculation is crucial for determining unknown lengths in a triangle, just like in the original exercise where the lengths of the sides of right triangles were calculated using known angles and the altitude.

Triangle Area Calculation

Calculating the area of a triangle might seem like a complex task, but it becomes straightforward when using the right approach. For any triangle, the area can be found if you know the base and the height. The fundamental formula for the area of a triangle is:\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]Let's see how this applies to our original exercise. In the problem, two right triangles form part of the larger triangle due to the altitude drawn. The base is simply the sum of the calculated sides of these two right triangles. The height is the altitude given, which is 5 meters. By applying the formula, evaluate the base and then multiply by the height to find the area. Finally, divide by 2 to get the total area. Using the sine function previously discussed allows us to determine and sum these side lengths accurately.

Angles in Triangles

Understanding angles in triangles is a crucial concept in geometry and trigonometry. Every triangle has three angles and the sum of these angles will always be \(180^\circ\). When you have a triangle with a right angle (\(90^\circ\)), the remaining two angles must add up to \(90^\circ\). This property helps simplify solving problems using trigonometric functions.Here's the rundown:

• The two angles given in the exercise with the altitude are \(36^\circ\) and \(42^\circ\).
• The right angle formed by the altitude results in right triangles useful for applying trigonometric principles.

By knowing two angles, the problem becomes more manageable. The third angle in any triangle can be calculated if two angles are known, although it's not always necessary to solve every problem. Taking advantage of these angular relationships helps apply trigonometry to find missing sides or calculate the area, just as done in the original step-by-step solution.