What is the Largest Angle of a Triangle with Sides 4 cm, 7 cm, and 9 cm?

When faced with determining the largest angle in a triangle given its sides, one of the most effective methods is to apply the Law of Cosines. This mathematical tool allows us to deduce the measure of any angle in a triangle when the lengths of all three sides are known. In this article, we will explore how to find the largest angle in a triangle with sides of 4 cm, 7 cm, and 9 cm using the Law of Cosines.

Understanding the Law of Cosines

The Law of Cosines, also known as the Cosine Rule, is a fundamental theorem in geometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is given as:

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

Where (a, b,) and (c) are the lengths of the sides of the triangle, and (C) is the angle opposite side (c).

Applying the Law of Cosines

In the given problem, we have a triangle with sides of 4 cm, 7 cm, and 9 cm. To find the largest angle, we need to identify the side that will be the longest, as the angle opposite it will be the largest. In this case, the longest side is 9 cm.

Let (a 4) cm Let (b 7) cm Let (c 9) cm

The angle opposite to side (c) (which is 9 cm in this case) is the largest angle, denoted as (C).

Calculating the Largest Angle

Using the Law of Cosines, we substitute the values into the formula:

(9^2 4^2 7^2 - 2 cdot 4 cdot 7 cdot cos(C))

Calculating the squares:

(81 16 49 - 56 cdot cos(C))

Simplifying the equation:

(81 65 - 56 cdot cos(C))

Rearranging the equation to solve for (cos(C)):

(81 - 65 -56 cdot cos(C))

(16 -56 cdot cos(C))

(cos(C) -frac{16}{56} -frac{2}{7})

To find the angle (C), we apply the inverse cosine function:

(C cos^{-1}left(-frac{2}{7}right))

Using a calculator, we get:

(C approx 108.21^circ)

Therefore, the largest angle in the triangle is approximately (108.21^circ).

Revisiting the Solution with Different Steps

For comparison, let's reattempt the problem and find the same angle using a slightly different approach to verify our results:

Using the formula for the Law of Cosines:

(9^2 7^2 4^2 - 2 cdot 7 cdot 4 cdot cos(A))

Calculating the squares:

(81 49 16 - 56 cdot cos(A))

Simplifying the equation:

(81 65 - 56 cdot cos(A))

Rearranging the equation to solve for (cos(A)):

(81 - 65 -56 cdot cos(A))

(16 -56 cdot cos(A))

(cos(A) -frac{16}{56} -frac{2}{7})

To find the angle (A), we apply the inverse cosine function:

(A cos^{-1}left(-frac{2}{7}right))

Using a calculator, we get:

(A approx 106.6^circ)

Thus, the largest angle in the triangle is approximately (106.6^circ).

Conclusion

By applying the Law of Cosines, we have determined that the largest angle in a triangle with sides of 4 cm, 7 cm, and 9 cm is approximately (108.21^circ) or (106.6^circ), confirming that the angle opposite the longest side (9 cm) is indeed the largest. This method provides a rigorous and accurate way to find the largest angle in a triangle when the side lengths are given.

References:

Law of Cosines (Cosine Rule) Trigonometric functions and their inverses