There is exactly one triangle the length of whose sides are integers in arithmetic progression and whose area is 156 square units. Find the perimeter of this triangle.
To find the perimeter of the triangle, we first need to find the lengths of its sides.
Let's assume that the three sides of the triangle are given by the terms a-d, a, a+d, where a is the second term and d is the common difference of the arithmetic progression.
We can use Heron's Formula to calculate the area of the triangle in terms of the sides. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is given by:
A = sqrt(s(s-a)(s-b)(s-c))
where s = (a + b + c)/2 is the semiperimeter of the triangle.
In this case, the area of the triangle is given as 156 square units. So, we can write:
156 = sqrt(s(s-a)(s-a-d)(s-a+d))
To make it easier to work with, let's square both sides of the above equation:
(156)^2 = s(s-a)(s-a-d)(s-a+d)
Now, let's solve for s^2:
s^2 = (a^2 - d^2)/4
Since we are looking for integer side lengths, the expression (a^2 - d^2) must be a perfect square.
There are several ways to proceed from here. One approach is to test different values of a and d to find a pair that satisfies the equation. You can start with small values and incrementally increase them until you find a solution.
Using this method, let's test some values of a and d:
For a = 10 and d = 2:
s^2 = (10^2 - 2^2)/4 = 96
Since 96 is not a perfect square, try different values of a and d until you find one that yields a perfect square.
Continue this process until you find a pair of values for a and d that make the right side of the equation a perfect square.
Once you find a pair (a, d) that satisfies the equation, you can substitute them back into the expression for s^2 and solve for s.
Once you have the value of s, you can calculate the lengths of the triangle's sides (a-d, a, a+d) and find their sum to determine the perimeter of the triangle.