Final answer:
The length of the third side of an isosceles triangle with two sides measuring 23 cm and an angle between them of 54° can be calculated using the Law of Cosines, resulting in a calculation involving the cosine of 54° and the square root of the resulting expression.
Step-by-step explanation:
To determine the length of the third side of an isosceles triangle with legs each measuring 23 cm and an angle between the legs of 54°, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle γ opposite side c, the relationship between the sides and the angle is given by c² = a² + b² - 2ab*cos(γ). In an isosceles triangle, the two legs (a and b) are equal in length, so the formula simplifies to c² = 2a² - 2a²*cos(γ), which can be further simplified to c² = 2a²*(1 - cos(γ)).
Substituting the given values into the formula, we have:
c² = 2*(23 cm)²*(1 - cos(54°))
= 2*(529 cm²)*(1 - cos(54°))
= 1058 cm²*(1 - cos(54°))
Now, calculate cos(54°) using a calculator and subtract that value from 1. Multiply the result by 1058 cm² and then take the square root of the entire expression to find the length of side c.
The formula for the Law of Cosines is derived from the Pythagorean theorem as it applies to non-right triangles. In a right triangle, the Pythagorean theorem directly relates the lengths of the legs to the hypotenuse as a² + b² = c², which is specific to right triangles only.