Segment Lengths Day 2

We know that if two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. So we have that

6*a = 9 * 3
6a = 27
a = 4.5

We know that if two secants share the same endpoint, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. So we have that

4*20 = x*(x + 11)
80 = x² + 11x
0 = x² + 11 - 80
0 = (x - 5)(x + 16)

So, either x = 16 or x = 5. But we know x*(x + 11) = 80, so if x was 16 we would have that 16*(16+11) = 80, but 16*(16+11) = 16*27 = 432, therefore we must have that x = 5.

We need to find both the values of a and b, so we know that if a secant and a tangent share a point outside the circle then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. We also know that if we have two chords that intersect inside a circle the the product of the segments of one chord equals the product of the segments of the other chord. So, first we can solve for a by saying

Steve works at an aquarium. They recently installed a cylindrical aquarium that is four floors high and Steve needs to install the cables for the filter. They will go from the floor to the top of the aquarium at the corner of the platforms that are at each level. In order to do this, Steve must make sure that there is enough room for the cables and the insulator that will surround them. Steve knows that the diameter of the aquarium is 45 feet, and that the distance from the corner of the platform to the point of tangency with the tank is 14 feet. He needs at least 3 feet between the corner and the aquarium to install the cables. What is the distance from the corner to the aquarium? Will he have enough room to install the cables?

We know that if a secant and a tangent share a point outside the circle then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. So we have that

14² = x*(x + 45)
196 = x² + 45x
0 = x² + 45x - 196
0 = (x - 4)(x + 49)

So we have that either x = 4 or x = 49. However we must have that x*(x + 45) = 196 and if x = 49 we have that 49*(49 + 45) = 49*94 = 4606. Therefore we must have that x = 4.
So the aquarium is 4 feet away from the corner of the platform, and since Steve only needs 3 feet to install his cables, he will have enough room.