To solve the problem of finding the area of a triangle with sides 13, 14, and 15, we can use Heron's Formula or Pythagoras Theorem (by calculating the height). Here's the step-by-step solution using both methods:

Method 1: Heron's Formula

Heron's Formula states that the area (A) of a triangle with sides (a, b, c) is:
[A = \sqrt{s(s-a)(s-b)(s-c)}]
where (s = \frac{a+b+c}{2}) (semi-perimeter).

1. Calculate the semi-perimeter:
   [s = \frac{13+14+15}{2} = 21]

2. Substitute into Heron's Formula:
   [A = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21×8×7×6}]

3. Simplify the product inside the square root:
   [21×6 = 126, \quad 8×7 =56 \quad \Rightarrow \quad 126×56 =7056]

4. Take the square root:
   [A = \sqrt{7056} = 84]

Method 2: Using Base and Height

Let the base be 14. Split the base into two segments (x) and (14-x). Using Pythagoras:
[x^2 + h^2 =13^2 \quad \text{and} \quad (14-x)^2 +h^2=15^2]

Subtract the first equation from the second:
[(14-x)^2 -x^2 =225-169 \quad \Rightarrow \quad 196-28x=56 \quad \Rightarrow \quad x=5]

Then, (h^2=13^2 -5^2=169-25=144 \quad \Rightarrow \quad h=12)

Area: (\frac{1}{2}×14×12=84)

Answer: (\boxed{84})

作者声明:本文包含人工智能生成内容。