The field is in a right triangle shape.The hypotenuse of this field is 6 metres and the peerimeter of this field is 14 metres.What is the area of the lawn?
a^2 + b^2 = 36
a + b + 6 = 14 so a = 8-b
64 - 16 b + 2 b^2 =36
2 b^2 - 16 b + 28 = 0
b^2 - 8 b + 14 = 0
b = [ 8 +/- sqrt (64 -56) ]/2
b = [ 8 +/- 2sqrt 2 ]/2
b = 4+/-sqrt 2
if b = 4+sqrt 2
a = 4 - sqrt 2
A = (1/2)ab
= (1/2)(16-2) = 7 m^2
To find the area of the lawn, we first need to determine the lengths of the other two sides of the right triangle.
Let's denote the two legs of the right triangle as 'a' and 'b'.
We know that the hypotenuse, represented by 'c', has a length of 6 meters.
Using the Pythagorean theorem, we can express the relationship between the sides of a right triangle as follows:
a^2 + b^2 = c^2
Substituting the given values, we have:
a^2 + b^2 = 6^2
a^2 + b^2 = 36
We also know that the perimeter of the field is 14 meters. The perimeter of a right triangle is the sum of the lengths of all its sides. So, we can express this relationship as:
a + b + c = 14
Substituting the given values, we have:
a + b + 6 = 14
a + b = 14 - 6
a + b = 8
Now, we have a system of equations:
a^2 + b^2 = 36 (Equation 1)
a + b = 8 (Equation 2)
Using Equation 2, we can express 'a' in terms of 'b':
a = 8 - b
Substituting this value in Equation 1, we get:
(8 - b)^2 + b^2 = 36
Solving this quadratic equation will give us the values of 'a' and 'b'. However, from the original problem, we can analyze the triangle's hypotenuse and perimeter to determine that the legs are both shorter than the hypotenuse, so they must be less than 6 meters. Since 6^2 = 36, we can deduce that 'b' must be less than 6.
Trying different values, we find that when b = 3 and a = 5 (or vice versa), both equations are satisfied:
(8 - 3)^2 + 3^2 = 36
5^2 + 3^2 = 36
So, the lengths of the legs are a = 5 meters and b = 3 meters.
Now, we can calculate the area of the lawn using the formula for the area of a right triangle:
Area = (1/2) * base * height
The base and height of the triangle are the two legs, 'a' and 'b'. Substituting the values we found, we have:
Area = (1/2) * 5 * 3
Area = 7.5 square meters
Therefore, the area of the lawn is 7.5 square meters.