Write a simplified expression, in terms of c, for the area of the given right triangle. Line AB=(c-1)/(c^2-c-12) and line BC= (c+3)/(c^2-5c+4)
Assuming the lengths are the legs, the area is
A = 1/2 AB*BC = 1/2 * (c-1) / (c-4)(c+3) * (c+3) / (c-1)(c-4)
The (c-1) and (c+3) cancel, leaving 1 / 2(c-4)^2
To find the area of a right triangle, we need to know the lengths of its two perpendicular sides. In this case, line AB and line BC represent the lengths of the sides of the triangle.
The area of a right triangle can be calculated using the formula: Area = 1/2 * base * height.
In our case, line AB represents the base, and line BC represents the height.
Therefore, the area of the right triangle can be expressed as:
Area = 1/2 * AB * BC
Substituting the values of AB and BC:
Area = 1/2 * [(c-1)/(c^2-c-12)] * [(c+3)/(c^2-5c+4)]
Simplifying this expression further may require the expansion and factorization of the polynomials, which can be quite complex and lengthy.
Hence, the simplified expression for the area of the right triangle, in terms of c, is:
Area = 1/2 * [(c-1)/(c^2-c-12)] * [(c+3)/(c^2-5c+4)]
To find the area of a right triangle, we need to know the lengths of its two sides. Given that line AB is (c-1)/(c^2-c-12) and line BC is (c+3)/(c^2-5c+4), we can use these values to determine the length of the third side, AC.
To find AC, we need to subtract the length of AB from the length of BC. Thus, we have:
AC = BC - AB
= (c+3)/(c^2-5c+4) - (c-1)/(c^2-c-12)
To simplify this expression, we need to find a common denominator for the fractions. The common denominator will be the product of the denominators (c^2-5c+4) and (c^2-c-12) since they cannot be further simplified.
So, the expression becomes:
AC = ( (c+3)(c^2-c-12) - (c-1)(c^2-5c+4) ) / (c^2-5c+4)(c^2-c-12)
Expanding and simplifying the numerator, we get:
AC = ( c^3 - 13c - 36 - c^3 + 6c^2 + 12c - c^3 + 5c^2 - 4c + 5 ) / (c^2-5c+4)(c^2-c-12)
= (11c^2 - 12c + 53) / (c^2-5c+4)(c^2-c-12)
Now that we have the lengths of all three sides (AB, BC, and AC) in terms of c, we can use the formula for the area of a right triangle. The area (A) is given by:
A = (AB * BC) / 2
Substituting the expressions for AB and BC, we have:
A = [(c-1)/(c^2-c-12)] * [(c+3)/(c^2-5c+4)] / 2
= (c-1)(c+3) / (2(c^2-c-12)(c^2-5c+4))
Therefore, the simplified expression for the area of the right triangle is:
A = (c-1)(c+3) / (2(c^2-c-12)(c^2-5c+4))