adjacent side (in a triangle)

Question: Adjacent Side (in a Triangle)

In geometry, a triangle is a polygon with three sides and three angles. One of the important concepts in triangles is the adjacent side. In this blog, we will explore what the adjacent side in a triangle is, its definition, formula, application, symbol, methods, and provide solved examples and practice problems.

Definition of Adjacent Side (in a Triangle)

In a triangle, the adjacent side refers to the side that is next to a given angle. It is the side that shares a common vertex with the angle but does not form the angle itself. The adjacent side is also known as the "next to" side or the "side adjacent to" an angle.

Knowledge Points of Adjacent Side (in a Triangle)

Understanding the concept of adjacent side in a triangle requires knowledge of the following:

1. Triangle: The basic understanding of a triangle as a polygon with three sides and three angles.
2. Angles: Familiarity with angles and their measurements.
3. Vertex: Understanding the concept of a vertex, which is the point where two sides of a triangle meet.

Formula for Adjacent Side (in a Triangle)

The formula to calculate the length of the adjacent side in a triangle depends on the given information. If we know the length of the hypotenuse and one of the acute angles, we can use trigonometric functions to find the adjacent side.

For example, if we know the length of the hypotenuse (c) and the measure of one of the acute angles (θ), we can use the cosine function to find the adjacent side (a):

a = c * cos(θ)

Application of the Adjacent Side Formula

The formula for the adjacent side in a triangle is commonly used in trigonometry and geometry. It helps in solving various real-life problems involving triangles, such as calculating distances, heights, or angles in navigation, engineering, and physics.

By knowing the length of the hypotenuse and one of the acute angles, we can determine the length of the adjacent side, which is crucial for many calculations and measurements.

Symbol for Adjacent Side (in a Triangle)

The symbol commonly used to represent the adjacent side in a triangle is 'a'. It is derived from the word "adjacent" and is widely recognized in mathematical notation.

Methods for Finding the Adjacent Side (in a Triangle)

Apart from using the formula mentioned above, there are other methods to find the adjacent side in a triangle:

1. Pythagorean Theorem: If the triangle is a right-angled triangle, we can use the Pythagorean theorem to find the adjacent side.
2. Trigonometric Ratios: By using trigonometric ratios such as sine, cosine, or tangent, we can find the adjacent side in non-right-angled triangles.

Solved Example on Adjacent Side (in a Triangle)

Let's consider a right-angled triangle with a hypotenuse of length 10 units and an acute angle of 30 degrees. We need to find the length of the adjacent side.

Using the formula mentioned earlier, we can calculate the adjacent side (a):

a = 10 * cos(30°)
a ≈ 8.66 units

Therefore, the length of the adjacent side in this triangle is approximately 8.66 units.

Practice Problems on Adjacent Side (in a Triangle)

1. In a right-angled triangle, the hypotenuse is 15 units, and the adjacent side is 12 units. Find the measure of the acute angle.
2. In a triangle, the adjacent side is 5 units, and the opposite side is 3 units. Find the length of the hypotenuse.
3. In a triangle, the adjacent side is 7 units, and the angle opposite to it is 45 degrees. Find the length of the hypotenuse.

FAQ on Adjacent Side (in a Triangle)

Q: What is the adjacent side in a triangle? A: The adjacent side in a triangle is the side that is next to a given angle. It shares a common vertex with the angle but does not form the angle itself.

Q: How do you find the length of the adjacent side in a triangle? A: The length of the adjacent side can be found using trigonometric functions, such as cosine, when the length of the hypotenuse and one of the acute angles are known.

Q: What is the symbol for the adjacent side in a triangle? A: The symbol commonly used to represent the adjacent side in a triangle is 'a'.

Q: Can the adjacent side be longer than the hypotenuse in a triangle? A: No, the adjacent side cannot be longer than the hypotenuse in a triangle. The hypotenuse is always the longest side in a right-angled triangle.

Q: Are there any other methods to find the adjacent side in a triangle? A: Yes, apart from using trigonometric functions, the Pythagorean theorem can be used to find the adjacent side in a right-angled triangle.

Understanding the concept of the adjacent side in a triangle is essential for solving various mathematical problems and real-life applications. By knowing the formula, methods, and symbol associated with the adjacent side, we can confidently tackle problems involving triangles.