To find the area of a triangle given the side lengths, you can use Heron's formula. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c and semi-perimeter s is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle.
In your case, the given information is:
a = 8 cm
b = 7 cm
c = 6 cm
The difference between the semi-perimeter (s) and the sides is:
s - a = 8 cm
s - b = 7 cm
s - c = 6 cm
To solve for the semi-perimeter (s), we can set up a system of equations using the given differences:
s - a = 8
s - b = 7
s - c = 6
Adding a to all sides of the first equation, b to all sides of the second equation, and c to all sides of the third equation, we get:
s = 8 + a
s = 7 + b
s = 6 + c
Since all three equations are equal to s, they are equal to each other:
8 + a = 7 + b = 6 + c
From this equation, we can find the value of s:
8 + a = 7 + b
a - b = -1
7 + b = 6 + c
b - c = -1
Substituting the value of b from the first equation into the second equation:
a - b = -1
a - (-1) = -1
a + 1 = -1
a = -1 - 1
a = -2
Substituting the value of a into the first equation:
a - b = -1
-2 - b = -1
b = -2 + 1
b = -1
Substituting the values of a and b into the third equation:
s = 6 + c
-2 = 6 + c
c = -2 - 6
c = -8
However, it seems that the values obtained for a, b, and c are negative, which is not possible for the side lengths of a triangle. Please double-check the given values or provide additional information if necessary.
Answers & Comments
Answer:
Step-by-step explanation:
To find the area of a triangle given the side lengths, you can use Heron's formula. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c and semi-perimeter s is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle.
In your case, the given information is:
a = 8 cm
b = 7 cm
c = 6 cm
The difference between the semi-perimeter (s) and the sides is:
s - a = 8 cm
s - b = 7 cm
s - c = 6 cm
To solve for the semi-perimeter (s), we can set up a system of equations using the given differences:
s - a = 8
s - b = 7
s - c = 6
Adding a to all sides of the first equation, b to all sides of the second equation, and c to all sides of the third equation, we get:
s = 8 + a
s = 7 + b
s = 6 + c
Since all three equations are equal to s, they are equal to each other:
8 + a = 7 + b = 6 + c
From this equation, we can find the value of s:
8 + a = 7 + b
a - b = -1
7 + b = 6 + c
b - c = -1
Substituting the value of b from the first equation into the second equation:
a - b = -1
a - (-1) = -1
a + 1 = -1
a = -1 - 1
a = -2
Substituting the value of a into the first equation:
a - b = -1
-2 - b = -1
b = -2 + 1
b = -1
Substituting the values of a and b into the third equation:
s = 6 + c
-2 = 6 + c
c = -2 - 6
c = -8
However, it seems that the values obtained for a, b, and c are negative, which is not possible for the side lengths of a triangle. Please double-check the given values or provide additional information if necessary.
Solution :
Given, the difference between semi perimeter s and the sides a b and c of triangle ABC are 8cm,7cm and 6cm respectively.
To find : Area of triangle ABC.
Here, semi perimeter is s.
According to the question :
→ s - a = 8 ..(1)
→ s - b = 7 ..(2)
→ s - c = 6 ..(3)
As, s = semi perimeter of triangle ABC.
→ s = (a + b + c)/2
→ a + b + c = 2s
Now adding all 3 equations :
→ s - a + s - b + s - c = 8 + 7 + 6
→ 3s - (a + b + c) = 21
→ 3s - 2s = 21
→ s = 21 cm.
Now using Heron's formula :
→ Area of ∆ = √[s(s - a)(s - b)(s - c)]
Substituting all values,
→ Area of ∆ABC = √[21 × 8 × 7 × 6] cm²
→ Area of ∆ABC = √(21 × 336) cm²
→ Area of ∆ABC = √(7056) cm²
→ Area of ∆ABC = 84 cm² (Ans.)