Finding the Area of a Triangle with Side Lengths of 5 cm, 6 cm, and 7 cm

When dealing with triangles and their properties, one of the most common tasks is calculating the area. In this case, we are interested in a triangle with sides of 5 cm, 6 cm, and 7 cm. Through various methods, we can determine the area of this triangle. Let's explore the different approaches and delve into the details of each step.

Approach 1: Using Right Angled Triangle Property

One immediate observation is the use of the Pythagorean theorem to identify if the triangle is a right-angled triangle. For a triangle with sides (a), (b), and (c), if (a^2 b^2 c^2), it is a right-angled triangle. Here, let's assume the sides are 5 cm, 6 cm, and 7 cm, and we need to check if it fits the Pythagorean theorem.

Step 1: Calculate (a^2 b^2): [5^2 6^2 25 36 61] br Step 2: Calculate (c^2): [7^2 49]

Since (25 36 eq 49), the triangle is not a right-angled triangle. This approach does not directly apply, so we need to use another method.

Approach 2: Using Heron's Formula

Step 1: Calculate the semiperimeter (s) of the triangle. [s frac{a b c}{2} frac{5 6 7}{2} 9text{ cm}] Step 2: Apply Heron's formula to find the area of the triangle. [text{Area} sqrt{s(s - a)(s - b)(s - c)}] Step 3: Substitute the values into the formula. [text{Area} sqrt{9(9 - 5)(9 - 6)(9 - 7)} sqrt{9 times 4 times 3 times 2} sqrt{216} 6sqrt{6}text{ cm}^2]

This approach is robust and works for any triangle, given the side lengths.

Approach 3: Simplified Heron's Formula Calculation

Step 1: Calculate the semiperimeter (s) again for simplicity. [s frac{5 6 7}{2} 9text{ cm}] Step 2: Use the simplified form of Heron's formula. [text{Area} sqrt{s(s - a)(s - b)(s - c)} sqrt{9(9 - 5)(9 - 6)(9 - 7)}] Step 3: Perform the multiplication inside the square root. [text{Area} sqrt{9 times 4 times 3 times 2} sqrt{216} 6sqrt{6}text{ cm}^2]

Both approaches confirm that the area of the triangle is approximately (14.7text{ cm}^2) when rounded to one decimal place. The exact value in radicals is (6sqrt{6}text{ cm}^2).

In conclusion, while recognizing whether a triangle is a right-angled triangle is useful for specific cases, for a general triangle, Heron's formula is the most reliable method. By calculating the semiperimeter and applying Heron's formula, the area of the triangle with side lengths 5 cm, 6 cm, and 7 cm is determined to be (6sqrt{6}text{ cm}^2).