Question

In the figure, TU is tangent to the circle at point U. Use the figure to answer the question

Answer 1

Explanation:

intersecting secants theorem :

let's assume the second line would not be a tangent but a true secant with a second circle intersection point V, then

ST × RT = TU × TV

in other words : the product of the lengths of the short (external) segment and the full secant is the same for every secant line originating from the same point.

now we have the extreme case that U = V due to the tangent situation.

but the same principle applies. just TU = TV, and therefore

ST × RT = TU²

RT = TU²/ST

for the same reason and with the same approach, yes, we can calculate TU out of a given set of RT and ST :

TU² = ST × RT

TU = sqrt(ST × RT) = sqrt(4 × 13) = sqrt(52) = 2×sqrt(13) =

= 7.211102551...

Answer 2

see explanation

Explanation:

given a tangent and a secant drawn from an external point to the circle, then the square of the measure of the tangent is equal to the product of the secant's external part and the entire secant , that is

ST × RT = TU²

given RT = 13 and ST = 4 , then

TU² = 4 × 13 = 52 ( take square root of both sides )

TU =
`√(52)` = 2
`√(13)` ≈ 7.21 ( to 2 decimal places )