ab and ac are tangent to P find ab

Question

possible answers:

ab= 2

ab= 11/2

ab= 1/2

ab=10

in the actual question, AC= 11y

AB= 3y+4

Answer 1

Two tangents to a circle from the same point outside the circle will always be equal. We set the two expressions equal to each other and solve for y.

11y = 3y + 4

8y = 4

y = 1/2

Once we've found that y = 1/2, we substitute that value back into the original expression for AB and solve.

AB = 3y + 4

AB = 3(1/2) + 4

AB = (3/2) + 4

AB = 11/2

Answer 2

If the two segments are tangent to the same point, then they are congruent (i.e., ab ~ ac).

You are given the following:

          ac = 11y          and          ab = 3y + 4

Since we've already determined that the two segments (ab and ac) and congruent, we can set the expressions that define their lengths equal to one another:

          11y = 3y + 4

Now we solve for the unknown variable (y) by first subtracting 3y from both sides of the equation then dividing both sides by the coefficient of y:

          11y - 3y = 3y - 3y + 4

                   8y = 4

               8y/8 = 4/8

                    y = 1/2

To find ab, we plug in the solution for y into the expression defining ab:

     ab = 3y + 4

     ab = 3(1/2) + 4

         = 3/2 + 4

Since one of the terms is a fraction, we need to find a common denominator among the 2 terms so that we can add them. To do so, multiply 4 by 2/2 to get 8/2.

     ab = 3/2 + 4(2/2)

         = 3/2 + 8/2

         = (3+8)/2

    ab = 11/2