Answer:

458.8 \, \text{cm}^2

Step-by-step explanation:

To find the area of a triangle when you have the lengths of two sides (a and b) and the perimeter (P), you can use Heron's formula. Heron's formula states that the area (A) of a triangle with sides of lengths a, b, and c (where c is the third side) and semiperimeter (s) is:

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]

Where:

- \(s\) is the semiperimeter, which is half of the perimeter (P/2).

- \(a\) and \(b\) are the given side lengths.

- \(c\) can be calculated as \(c = P - a - b\).

In your case:

- \(a = 60\) cm

- \(b = 40\) cm

- \(P = 156\) cm

Now, calculate the semiperimeter (s):

\[s = \frac{P}{2} = \frac{156}{2} = 78\text{ cm}\]

Next, calculate \(c\):

\[c = P - a - b = 156 - 60 - 40 = 56\text{ cm}\]

Now, you have all the values needed to calculate the area (A) using Heron's formula:

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]

\[A = \sqrt{78(78-60)(78-40)(78-56)}\]

\[A = \sqrt{78 \cdot 18 \cdot 38 \cdot 22}\]

\[A = \sqrt{210492}\]

Now, calculate the square root:

\[A ≈ 458.8\text{ cm}^2\]

So, the area of the triangle is approximately \(458.8 \, \text{cm}^2\).