Construct a circle with radius 4 cm. Draw a diameter PT. Construct a tangent at T. Mark a point R on the tangent such that TR = 9 cm. Calculate the length of PR.

• Mark a point O as the center.
• Using a ruler, open your compasses to $4 \text{ cm}$.
• Place the compass point at O and draw a circle.

• Draw a straight line passing through O. This line will intersect the circle at two points. Label these points P and T. PT is a diameter.
• To construct the tangent at T:
  • Extend the line PT beyond T.
  • Place the compass point at T and draw two arcs of equal radius that intersect the extended line PT on either side of T. Let these intersection points be X and Y.
  • Open your compasses to a radius greater than TX. Place the compass point at X and draw an arc.
  • With the same compass opening, place the compass point at Y and draw another arc that intersects the first arc. Let this intersection point be Z.
  • Draw a straight line through T and Z. This line TZ is the tangent to the circle at T.

• Place the compass point at T.
• Using a ruler, open your compasses to $9 \text{ cm}$.
• Draw an arc on the tangent line TZ. Label the intersection point R.

• Draw a straight line connecting point P to point R.
• Measure the length of PR with a ruler.

Since this is a digital environment, I will calculate the length of PR.
The radius of the circle is $OT = 4 \text{ cm}$.
Since PT is a diameter, $PT = 2 \times OT = 2 \times 4 \text{ cm} = 8 \text{ cm}$.
The tangent at T is perpendicular to the diameter PT at T. Therefore, $\triangle PTR$ is a right-angled triangle with the right angle at T.
We have $PT = 8 \text{ cm}$ and $TR = 9 \text{ cm}$.
Using the Pythagorean theorem:

As a decimal, $PR \approx 12.04 \text{ cm}$ (to 2 decimal places).

The length of PR is $\sqrt{145} \text{ cm} \approx 12.0 \text{ cm}$