There are two noncongruent triangles that have a 9-inch side, a 10-inch side, and that enclose 36 square inches of area. Find the length of the third side in each of these triangles.
To find the length of the third side in each of the triangles, we can use the Heron's formula. Heron's formula relates the lengths of the sides of a triangle to its area.
Let's assume the third side of the first triangle is denoted as x inches. According to Heron's formula, the area of the triangle is given by:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides.
In this case, the area is given as 36 square inches, and the lengths of two sides are 9 inches and 10 inches. The semi-perimeter can be calculated as:
s = (9 + 10 + x)/2 = (19 + x)/2
Substituting these values into the formula, we have:
36 = sqrt((19 + x)/2 * ((19 + x)/2 - 9) * ((19 + x)/2 - 10) * ((19 + x)/2 - x))
Squaring both sides of the equation to eliminate the square root, we get:
36^2 = ((19 + x)/2 * ((19 + x)/2 - 9) * ((19 + x)/2 - 10) * ((19 + x)/2 - x))
Expanding the equation, we have:
1296 = ((19 + x)/2 * (10 - x/2) * (9 - x/2) * (19 - x/2))
Simplifying further, we can multiply the terms on the right side:
1296 = ((19 + x)/2 * (19 - x/2)^2 * (10 - x/2))
Dividing both sides by ((19 + x)/2 * (19 - x/2)^2), we get:
1296 / ((19 + x)/2 * (19 - x/2)^2) = 10 - x/2
Now, we can solve this equation to find the value of x.