### 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 a formula that involves the side lengths and some trigonometry.
Since this triangle has two equal sides of length 9, it is an isosceles triangle, and I can draw an altitude from the vertex opposite the base of length 8 to the midpoint of the base.
This altitude divides the triangle into two right triangles, each with hypotenuse 9 and base 4, since the midpoint of the base is 4 units away from either endpoint.
I can use the Pythagorean theorem to find the length of the altitude, which is also the height of the original triangle.
Let h be the length of the altitude.
Then, by the Pythagorean theorem, I have h^2 + 4^2 = 9^2, or h^2 = 9^2 - 4^2 = 65.
Taking the square root of both sides, I get h = sqrt(65), which is an irrational number.
Now, I can use the formula for the area of a triangle, which is A = (1/2)bh, where b is the base and h is the height.
Plugging in the values I have, I get A = (1/2)(8)(sqrt(65)), which is the exact answer.
If I want to approximate the answer to a decimal, I can use a calculator to find that sqrt(65) is about 8.06.
Then, A = (1/2)(8)(8.06) = 32.24, which I can round to two decimal places.