Final answer:
In an equilateral triangle with sides 7y + 19 and 11y - 89, the value of y is determined to be 27. Substituting y, the length of side c is calculated to be 208. Since all angles in an equilateral triangle are 60 degrees, the angle expression 8x - 44 equals 60, yielding x as 13.
Step-by-step explanation:
Finding Values of c and y in an Equilateral Triangle
In an equilateral triangle, all sides have the same length and all angles are equal. This means that each angle has a measure of 60 degrees. If we have an equilateral triangle ABC with sides AB and BC, given as AB = 7y + 19 and BC = 11y - 89, we can find the value of y by equating these expressions, since they represent the lengths of the sides in an equilateral triangle.
Therefore, we can write the equation:
By solving this equation, we can find the value of y. Moving the y terms to one side and the constants to the other, we get:
- 7y - 11y = -89 - 19
- -4y = -108
- y = 27
Now, to find the value of c, we can substitute y into one of the side expressions:
- c = 7y + 19
- c = 7(27) + 19
- c = 189 + 19
- c = 208
For the angle C, or angle c as mentioned in the question, which is given by 8x - 44, we know that in an equilateral triangle, all angles are 60 degrees. Therefore,
- 8x - 44 = 60
- 8x = 104
- x = 13
With these calculations, we have found that the value of y is 27, the length c is 208, and the value of x is 13.