Find the area of a triangle whose sides are 13 cm, 14 cm and 15 cm, using Heron's formula.
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Step-by-Step Solution
Step 1: Calculate the semi-perimeter
Heron's formula requires the semi-perimeter of the triangle, denoted by s. The semi-perimeter is half the sum of the lengths of the three sides of the triangle. Here, the side lengths are a=13 cm, b=14 cm, and c=15 cm.
Step 2: Substitute values and compute semi-perimeter
Substitute the given side lengths into the formula for the semi-perimeter and perform the addition and division to find its value. The sum of the sides is 13+14+15=42 cm, and half of this sum is 21 cm.
Step 3: Apply Heron's formula
Heron's formula provides a way to calculate the area of a triangle when only the lengths of its three sides are known. The formula involves the semi-perimeter s and the lengths of the sides a, b, and c.
Step 4: Substitute values into Heron's formula
Substitute the calculated semi-perimeter s=21 cm and the given side lengths a=13 cm, b=14 cm, c=15 cm into Heron's formula. This sets up the expression for the area calculation.
Step 5: Simplify the expression
Perform the subtractions inside the parentheses: 21−13=8, 21−14=7, and 21−15=6. This simplifies the expression under the square root.
Step 6: Calculate the product and square root
Multiply the numbers under the square root: 21×8×7×6=7056. Then, calculate the square root of 7056, which is 84. The area is expressed in square centimeters.