SOLUTION: find the area of the triangle with vertices A(-1,3)B(2,3)C(2,6)

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Question 461705: find the area of the triangle with vertices A(-1,3)B(2,3)C(2,6)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
the area of the triangle with vertices
A(-1,3)
B(2,3)
C(2,6)
first find a distance between vertices which is equal to the length of sides:
a distance between vertices
A(-1,3)
B(2,3)
is:
Solved by pluggable solver: Distance Formula to determine length on coordinate plane
The distance (d) between two points is given by the following formula:

d=sqrt%28%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2%29

Thus in our case, the required distance is
d=sqrt%28%282--1%29%5E2+%2B+%283-3%29%5E2%29=+3+


For more on this concept, refer to Distance formula.


so, AB=c=3
a distance between vertices
A(-1,3)
C(2,6)
is:
Solved by pluggable solver: Distance Formula to determine length on coordinate plane
The distance (d) between two points is given by the following formula:

d=sqrt%28%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2%29

Thus in our case, the required distance is
d=sqrt%28%282--1%29%5E2+%2B+%286-3%29%5E2%29=+4.24264068711928+


For more on this concept, refer to Distance formula.


so, AC=b=4.24
a distance between vertices
B(2,3)
C(2,6)
is:
Solved by pluggable solver: Distance Formula to determine length on coordinate plane
The distance (d) between two points is given by the following formula:

d=sqrt%28%28x2-x1%29%5E2+%2B+%28y2-y1%29%5E2%29

Thus in our case, the required distance is
d=sqrt%28%282-2%29%5E2+%2B+%286-3%29%5E2%29=+3+


For more on this concept, refer to Distance formula.


so, BC=a=3

Solved by pluggable solver: Hero's (or Heron's) Formula (Used to Find the Area of a Triangle Given its Three Sides)


In order to find the area of a triangle 'A' with side lengths of 'a', 'b', and 'c', we can use Hero's Formula:



A=sqrt%28S%28S-a%29%28S-b%29%28S-c%29%29 where S is the semiperimeter and it is defined by S=%28a%2Bb%2Bc%29%2F2

Note: "semi" means half. So the semiperimeter is half the perimeter.



So let's first calculate the semiperimeter S:



S=%28a%2Bb%2Bc%29%2F2 Start with the semiperimeter formula.



S=%283%2B4.24%2B3%29%2F2 Plug in a=3, b=4.24, and c=3.



S=%2810.24%29%2F2 Add.



S=5.12 Divide.



----------------------------------------



A=sqrt%28S%28S-a%29%28S-b%29%28S-c%29%29 Now move onto Hero's Formula.



A=sqrt%285.12%285.12-3%29%285.12-4.24%29%285.12-3%29%29 Plug in S=5.12, a=3, b=4.24, and c=3.



A=sqrt%285.12%282.12%29%280.88%29%282.12%29%29 Subtract.



A=sqrt%2820.24996864%29 Multiply.



A=4.49999651555421 Take the square root of 20.24996864 to get 4.49999651555421.



So the area of the triangle with side lengths of a=3, b=4.24, and c=3 is roughly 4.49999651555421 square units.