A tangent is drawn from a point P to a circle, which meets the circle at T such that PT = 10.5 cm. A secant PAB intersects the circle in points A and B. If PA = 7 cm. What is the length (in cm) of the chord AB?

To find the length of the chord AB, we can apply the Tangent-Secant Theorem for a circle.

The Tangent-Secant Theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the external secant segment and the entire secant segment.

In this problem:
- The point P is the exterior point.
- PT is the tangent segment to the circle at point T, with PT = 10.5 cm.
- PAB is the secant line intersecting the circle at points A and B, where PA is the external secant segment and PB is the entire secant segment. We are given PA = 7 cm.

Let the length of the chord AB be represented by x cm.
The entire length of the secant segment PB is:
PB = PA + AB = 7 + x

According to the Tangent-Secant Theorem:
${\mathrm{PT}}^2=\mathrm{PA}·\mathrm{PB}$

Now, substitute the given values into the equation:
${\left(10.5\right)}^2=7·(7+x)$

Calculate the square of 10.5:
$110.25=7·(7+x)$

Divide both sides of the equation by 7 to solve for (7 + x):
$7+x=\frac{110.25}{7}$
$7+x=15.75$

Subtract 7 from both sides to find the value of x:
$x=15.75-7$
$x=8.75$

Thus, the length of the chord AB is 8.75 cm.