a traingle has an area 48 square cm and its shorter sides have lengths 9cm and 12cm. find the lagest angle of the traingle.

Area of triangle = side1*side2*sin(angle between them)/2

so 48 = 1/2(12)(9)sin ß
sin ß = 48/54
ß = 62.7º or 117.26º

If the angle is 62.7
Using the cosine law I found the side across from the 62.7º to be 11.226 and then using the sine law once more found the angle across from the side of 9 to be 45.44º surely making all three angles acute.

So the largest angle in the triangle must be in the second case, where the angle was 117.7º

To find the largest angle of a triangle, we first need to determine the lengths of all three sides of the triangle.

Given that the shorter sides have lengths 9cm and 12cm, we can use the formula for the area of a triangle to find the length of the third side.

The area of a triangle is given by the formula:
Area = (base * height) / 2.

In this case, the two sides of the triangle, 9cm and 12cm, form the base and height of the triangle. Substituting the given values into the formula, we can solve for the area:
48 = (9 * 12) / 2.

Multiplying both sides of the equation by 2:
96 = 9 * 12.

Dividing both sides of the equation by 9, we get:
96 / 9 = 12.

Therefore, the missing side of the triangle has a length of 12cm.

To find the largest angle, we can use the Law of Cosines. The formula is as follows:

c^2 = a^2 + b^2 - 2ab * cos(C),

where a, b, and c are the lengths of the sides of the triangle and C is the largest angle.

In this case, the lengths of the sides of the triangle are 9cm, 12cm, and 12cm. Substituting the values into the formula, we can solve for cos(C):

12^2 = 9^2 + 12^2 - 2*9*12*cos(C).

Simplifying the equation:
144 = 81 + 144 - 216*cos(C),
144 = 225 - 216*cos(C).

Rearranging the equation and dividing both sides by 216:
216*cos(C) = 225 - 144,
216*cos(C) = 81.

Dividing both sides by 216:
cos(C) = 81 / 216.

Taking the inverse cosine (cos^-1) of both sides to find the angle C:
C = cos^-1(81 / 216).

Using a calculator, we get:
C ≈ 55.68 degrees.

Therefore, the largest angle of the triangle is approximately 55.68 degrees.