The length of a side AB and side AC of an isosceles triangle ABC are 12cm and 8cm respectively. The size of an amgle C is 59°.Find the length of side AC?

Hmmm. I'd say that AC=8

Also, the 3rd side is either 8 or 12, right?

To find the length of side AC in the given isosceles triangle ABC, we can use the sine rule. The sine rule states that in any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this can be written as:

a / sin(A) = b / sin(B) = c / sin(C)

In our case, we are given the lengths of side AB and side AC, as well as the size of angle C. We need to find the length of side AC.

Let's assign variables:

a = 12cm (length of side AB)
b = AC (length of side AC)
C = 59° (angle C)

We are looking for the value of b (AC), so we can rewrite the sine rule as:

12 / sin(A) = b / sin(59°)

To find the value of sin(A), we can use the fact that the sum of angles in a triangle is 180°. Since angle A is equal to angle C in an isosceles triangle, we can write:

A + C + C = 180°
A + 59° + 59° = 180°
A = 180° - 118°
A = 62°

Using the sine rule again, we substitute the values and solve for b:

12 / sin(62°) = b / sin(59°)

To find the value of b, we can cross multiply:

12 * sin(59°) = b * sin(62°)

Now, divide both sides by sin(62°) to isolate b:

b = (12 * sin(59°)) / sin(62°)

Using a calculator, we can compute sin(59°) and sin(62°):

sin(59°) ≈ 0.857
sin(62°) ≈ 0.883

Substituting these values, we find:

b ≈ (12 * 0.857) / 0.883
b ≈ 11.65 cm

Therefore, the length of side AC is approximately 11.65 cm.