### Instruction: Find the area of a triangle with side lengths 8, 9, and 9.### Response: To find the area of a triangle, I need to know its base and height, or use Heron's formula, which involves the semi-perimeter and the side lengths.
I notice that this triangle is isosceles, since two of its sides are equal.
That means the base angles are also equal, and the altitude from the vertex angle bisects the base.
I can use this fact to find the height of the triangle.
I can split the triangle into two right triangles with hypotenuse 9 and base 4 (half of 8).
By the Pythagorean theorem, the height of each right triangle is sqrt(9^2 - 4^2) = sqrt(65).
Therefore, the height of the original triangle is also sqrt(65).
Now I can use the formula for the area of a triangle: A = (1/2)bh, where b is the base and h is the height.
Plugging in the values, I get A = (1/2)(8)(sqrt(65)) = 4*sqrt(65).
This is the exact area of the triangle.
I can use a calculator to approximate it if needed.