Answer:
x = 9
PR = 10.4
PT = 13
Explanation:
Pre-Solving
We are given a circle, with secants PR and PT. Secant PR is made up of PQ and QR, while Secant PT is made up of PS and ST.
We know that PQ = 5, QR = 5.4, PS = 4, and ST = x
As stated by your problem, we want to find the length of x (which is ST) and the length of each segment (i.e. we want to find the lengths of both PR and PT).
Solving
You're correct in setting up the Secant-Secant Product Theorem; it will be PR * PQ = PT * PS
Now, we substitute the values into the equation.
As stated above, PR is made up of PQ and QR, which means that PR = PQ + QR (segment addition postulate).
Substituting in the values, PR = 5 + 5.4 = 10.4
We also know that PT is made up of PS and ST. In other words, PT = PS + ST (also segment addition postulate).
Substituting in the values, PT = 4 + x
We already know the values of PQ and PS, so when we substitute the values in, the the equation becomes:
10.4 * 5 = (4 + x) * 4
We can simplify this to become:
52 = 16 + 4x
Subtract 16 from both sides.
36 = 4x
Divide both sides by 4.
9 = x
We found the value of x, however we aren't done yet. Remember that we need to find the lengths of the secants as well.
We have already found the length of PR (which was 10.4), but remember that PT was 4 + x
Substituting 9 as x, PT is 13.