a) FB = √(S² + (3b)²) b) FD = √((4a)² + (3b)²) c) CB = √(2S² + 9b²)
a) To express FB in terms of b, we need to find the length of BC. Since BC is the base of an equilateral triangle, its length is equal to the side length of the triangle. Let's call the side length of the equilateral triangle S. Then, we have BC = S. Since BC is one side of triangle BCF and FC is another side, we can use the Pythagorean theorem to find the length of BF.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Applying this theorem to triangle BFC, we have BF² = BC² + FC². Substituting the values we know, BF² = S² + (3b)². Taking the square root of both sides, we get BF = √(S² + (3b)²).
b) To express FD in terms of a and b, we can use a similar approach as in part (a). Since FD is one side of triangle FDE and DE is another side, we can use the Pythagorean theorem to find the length of FD.
Applying the theorem to triangle FDE, we have FD² = FE² + DE². Substituting the values we know, FD² = (4a)² + (3b)². Taking the square root of both sides, we get FD = √((4a)² + (3b)²).
c) To express CB in terms of a and b, we can use a similar approach as in part (a). Since CB is one side of triangle BCF and FC is another side, we can use the Pythagorean theorem to find the length of CB. Applying the theorem to triangle BCF, we have CB² = BC² + BF².
Substituting the values we know, CB² = S² + (√(S² + (3b)²))². Simplifying the expression, CB² = S² + S² + 9b². Combining like terms, CB² = 2S² + 9b². Taking the square root of both sides, we get CB = √(2S² + 9b²).