Question

$\overline{su}$ and $\overline{vt}$ are chords that intersect at point r.
what is the length of line segment vt?
4 units
8 units
13 units
14 units
the lengths of the segments are: $sr = x+6$, $ru = x$, $vr = x+1$, $rt = x+4$

Explanation:

Step1: Apply intersecting chords theorem

When two chords intersect in a circle, the products of the lengths of their segments are equal:
$$(x+1)(x+4) = (x+6)x$$

Step2: Expand both sides of equation

$$x^2 + 4x + x + 4 = x^2 + 6x$$

Step3: Simplify and solve for x

Combine like terms and isolate x:
$$x^2 + 5x + 4 = x^2 + 6x$$
$$4 = x$$

Step4: Calculate length of VT

VT is the sum of its segments:
$$VT = (x+1) + (x+4)$$
Substitute $x=4$:
$$VT = (4+1) + (4+4) = 5 + 8 = 13$$

Answer:

13 units