For painting a triangular courtyard and a square courtyard, an equal amount of paint is required. if the edges of the triangular courtyard measure 16.5 m, 27.5 m and 22 m, find the length of the square courtyard, correct up to 2 decimal places.
https://www.mathsisfun.com/geometry/herons-formula.html
13.47 m
Thank you.
To find the length of the square courtyard, we can start by calculating the area of both the triangular courtyard and the square courtyard.
1. Area of the triangular courtyard:
To find the area of a triangle, we can use Heron's formula. The formula is given by:
Area = sqrt(s × (s - a) × (s - b) × (s - c))
where s is the semi-perimeter of the triangle and a, b, and c are the lengths of its sides.
In this case, the lengths of the sides are given as 16.5 m, 27.5 m, and 22 m. We can calculate the semi-perimeter using the formula:
s = (a + b + c) / 2
s = (16.5 + 27.5 + 22) / 2
s = 33.25 m
Now, we can substitute the values into the area formula:
Area = sqrt(33.25 × (33.25 - 16.5) × (33.25 - 27.5) × (33.25 - 22))
Area = sqrt(33.25 × 16.75 × 5.75 × 11.25)
Area ≈ 269.50 square meters (rounded to 2 decimal places)
2. Area of the square courtyard:
Let's assume the side length of the square courtyard as 'x' meters.
The area of a square is given by:
Area = Side length × Side length
Area = x × x
Area = x^2
According to the given information, the area of the triangular courtyard and the square courtyard is equal. Therefore, we can equate the two areas and solve for 'x':
x^2 = 269.50
To find the value of 'x', we can take the square root of both sides:
x = sqrt(269.50)
x ≈ 16.42 meters (rounded to 2 decimal places)
Therefore, the length of the square courtyard is approximately 16.42 meters.