Problem 44 - Trigonometry

Question

Find the semiperimeter of triangle \(A B C\) with \(a=17, b=41\), and \(c=28\). a. 86 b. 172 C. 43 d. 29

Step-by-Step Solution

Answer

The semiperimeter of triangle \(ABC\) is 43.

Step 1: Understand the Semiperimeter Formula

The semiperimeter of a triangle is found by taking the sum of the lengths of its sides and then dividing by 2. The formula for the semiperimeter \( s \) is: \[ s = \frac{a + b + c}{2} \]This formula gives us half of the perimeter of the triangle.

Step 2: Substitute Given Side Lengths into the Formula

Given the side lengths \( a = 17 \), \( b = 41 \), and \( c = 28 \), substitute these values into the semiperimeter formula:\[ s = \frac{17 + 41 + 28}{2} \]

Step 3: Perform the Addition in the Numerator

Calculate the sum of the side lengths:\[ 17 + 41 + 28 = 86 \]Now our equation looks like this:\[ s = \frac{86}{2} \]

Step 4: Divide the Sum by 2

Finally, divide the total sum by 2 to find the semiperimeter:\[ s = \frac{86}{2} = 43 \]

Key Concepts

Understanding Triangles

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle's interior angles always sum up to 180 degrees. Triangles can vary in size and shape, and they come in different types based on their side lengths and angles. The three main types are:

• Equilateral Triangle: All three sides and angles are equal, with each angle measuring 60 degrees.
• Isosceles Triangle: Two sides are equal, and the angles opposite these sides are equal as well.
• Scalene Triangle: All sides and angles are different.

Additionally, triangles can be classified by their angles as acute, obtuse, or right triangles. A right triangle has one 90-degree angle, while obtuse triangles have one angle greater than 90 degrees. Acute triangles have all angles less than 90 degrees.
Understanding these basic classifications of triangles helps in solving various mathematical problems, including those involving the calculation of perimeters and areas.

Exploring the Semiperimeter Formula

The semiperimeter formula is a key concept in triangle geometry. It helps in simplifying calculations, especially when determining areas using Heron’s formula. The semiperimeter (denoted as \( s \)) is essentially half of the triangle's perimeter.
To calculate the semiperimeter of a triangle, add up the lengths of all three sides \( (a, b, c) \) and divide the result by 2. The formula looks like this:

• \( s = \frac{a + b + c}{2} \)

Here, \( a, b, \) and \( c \) represent the lengths of the triangle's sides.
This concept is quite useful because it is often easier or necessary to work with half of the perimeter in some formulas, including those used in finding the area when the side lengths are known. For example, Heron's formula, which is used for finding the area of a triangle with known side lengths, requires the semiperimeter as an integral part of the calculation.

Perimeter of a Triangle

The perimeter of a triangle is the total length around the triangle, calculated by summing the lengths of its three sides. The concept is straightforward but essential in geometry and everyday applications.
The formula for the perimeter \( P \) of a triangle is simply:

• \( P = a + b + c \)

Where \( a, b, \) and \( c \) are the lengths of the sides. Perimeter is a linear measure and is typically expressed in units such as meters, centimeters, or any other length unit appropriate to the problem.
Understanding the perimeter is crucial, as it helps in understanding the boundary of a shape or area and is also foundational for other concepts in geometry, such as area calculation and the semiperimeter concept mentioned earlier. Using the perimeter, one can easily move to find the semiperimeter by dividing the total by 2, which is often needed in various geometric calculations.