Solving a Triangle: Finding All Measures Given Three Sides

When the lengths of all three sides of a triangle are known, it becomes possible to determine all the other measures of the triangle, including the angles and the area. This article will guide you through the processes using the Law of Cosines and the Law of Sines. Understanding these methods can greatly enhance your ability to solve geometric problems efficiently.

Using the Law of Cosines

The Law of Cosines is a powerful tool in trigonometry that allows us to find the angles of a triangle when the lengths of all three sides are known. The formula for the Law of Cosines is:

(c^2 a^2 b^2 - 2ab cdot cos C)

Let's break down the steps:

Identify the sides of the triangle as (a), (b), and (c), with (c) being the side opposite to angle (C).

Rearrange the formula to solve for the cosine of the angle:

(cos C frac{a^2 b^2 - c^2}{2ab})

Once you have the cosine value, you can find the angles (A) and (B) using similar steps.

Example: Given a triangle with sides 6 cm, 8 cm, and 10 cm:

Apply the Law of Cosines to find (cos A):

(cos A frac{b^2 c^2 - a^2}{2bc} frac{8^2 10^2 - 6^2}{2 cdot 8 cdot 10} frac{64 100 - 36}{160} frac{128}{160} 0.8)

Calculate (A):

(A arccos(0.8) 36.87989765) degrees or 36.9 degrees

Find (B) using the Law of Sines:

(frac{a}{sin A} frac{b}{sin B} frac{c}{sin C})

Determine (C):

(C 180 - A - B 180 - 36.87989765 - 90 53.13010235) degrees or 53.1 degrees

Using the Law of Sines

The Law of Sines is another valuable technique for solving triangles. Once you have one angle, you can find the other angles using the following formula:

(frac{a}{sin A} frac{b}{sin B} frac{c}{sin C})

Let's see how to apply this to the example above:

From the previous step, you have (cos A 0.8). Calculate (A):

(A arccos(0.8) 36.87989765) degrees or 36.9 degrees

Using the Law of Sines to find (B):

(frac{8}{sin 36.9} frac{10}{sin B})

Solve for (sin B):

(sin B frac{10 sin 36.9}{8} 0.6)

Determine (B):

(B arcsin(0.6) 36.87989765) degrees or 36.9 degrees

Find (C):

(C 180 - A - B 180 - 90 - 36.9 53.1) degrees

Calculating the Area and Perimeter

When you have the lengths of all three sides, you can also calculate the area and the perimeter of the triangle:

Area Using Heron's Formula

Heron's formula is useful for finding the area of a triangle when the lengths of all three sides are known:

(s frac{a b c}{2})

(text{Area} sqrt{s(s-a)(s-b)(s-c)})

Example: For a triangle with sides 6 cm, 8 cm, and 10 cm:

Calculate the semi-perimeter:

(s frac{6 8 10}{2} 12)

Calculate the area:

(text{Area} sqrt{12(12-6)(12-8)(12-10)} sqrt{12 times 6 times 4 times 2} sqrt{576} 24 text{ square centimeters})

Perimeter

The perimeter of the triangle is simply the sum of the lengths of its sides:

(text{Perimeter} a b c 6 8 10 24 text{ centimeters})

By knowing the lengths of all three sides, you can determine the angles, the area, and the perimeter of a triangle, making the triangle fully defined and solvable. The combination of the Law of Cosines and the Law of Sines, along with Heron's formula, provides a robust set of tools for tackling a wide range of geometric problems.