Problem 4 - Algebra 2

Question

A triangle has sides of lengths 10 \(\mathrm{cm}\) and \(16 \mathrm{cm},\) and the measure of the angle between them is \(130^{\circ} .\) Find the area of the triangle.

Step-by-Step Solution

Answer

To find the area of the triangle, substitute the given dimensions (side length and included angle) into the formula and perform the calculations. The area is obtained in square cm.

Step 1: Identify the given dimensions

According to the exercise, the lengths of two sides of the triangle are \(10 cm\) and \(16 cm\). The included angle is \(130^{\circ}\).

Step 2: Apply the formula to calculate area

The formula for the area of a triangle in this case is \( \frac{1}{2}\times a \times b \times \sin C\). Substituting the given dimensions into the formula, we get \( Area = \frac{1}{2} \times 10 cm \times 16 cm \times \sin 130^{\circ}\).

Step 3: Calculate the area

After the substitution, perform the calculations to get the area of the triangle.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles. It provides tools and techniques to solve problems involving measurements where direct calculation isn't possible. In the context of this exercise, trigonometry is utilized to find the area of a triangle using the \(\sin\) function to relate an angle with its opposite side.

This problem involves a specific trigonometric formula for finding the area of a triangle when two sides and the included angle are known. The formula \(rac{1}{2} \times a \times b \times \sin C\) capitalizes on the relationship between the angle in the question and the sides surrounding it.

• \(a\) and \(b\) are the lengths of the two sides.
• \(C\) is the angle between these two sides.

Understanding trigonometry and its functions, like \(\sin\), helps solve problems where direct measurement or simple geometry fails, especially in non-right-angled triangles as seen in this exercise.

It’s key to remember how an angle's sine relates to these calculations, defining the height of the triangle if one side is visualized as the base. For obtuse angles, like \(130^{\circ}\), calculations still succeed by using this sine function effectively.

Law of Sines

The Law of Sines is pivotal in solving problems involving triangles, especially when angles or sides are needed. Though this exercise primarily focuses on area calculation, understanding how the Law of Sines could assist in further analysis of the triangle is essential. It states:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
where \(a, b, c\) are the triangle's sides opposite the angles \(A, B, C\) respectively.

• This law helps determine unknown angles or sides when partial information is available.
• It finds use in situations where the traditional Pythagorean theorem isn’t applicable, such as non-right triangles.

In our scenario with given sides and an included angle, the Law of Sines could extend calculations by helping identify unknown dimensions.

Imagine using it to find the missing angle, contributing to a full understanding of the triangle’s properties. Knowing the law ensures you can bridge angles and sides seamlessly, laying the groundwork for tackling more complex triangle-related problems.

Triangle Geometry

Triangle geometry involves exploring the properties and measures dealing with various types of triangles, such as scalene, isosceles, or equilateral. Each triangle embodies unique characteristics that dictate how calculations, like those for area, are performed. In this exercise, we're dealing with a potential scalene triangle based on the differing side lengths and angle.

Understanding geometry can aid in visualizing the setup of a triangle from its components - sides and angles.
This translates mathematical data into visual comprehensions using:

• Designated points as vertices.
• Segments as sides.
• Measurement of angles, crucial for distinction, especially exceeding \(90^{\circ}\) such as our \(130^{\circ}\) angle.

Familiarity with these geometric concepts can enhance your ability to measure, visualize, and solve area calculations or predict another geometric property. This awareness also prepares you for more intricate operations, like finding missing elements or discovering symmetries in part or the whole of a triangle, using observed properties comprehensively.