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.

Here are the steps for the construction: Step 1: Construct a circle. • Mark a point O as the center. • Using a ruler, open your compasses to 4 cm. • Place the compass point at O and draw a circle. Step 2: Construct a diameter and a tangent. • 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. Step 3: Locate point R. • Place the compass point at T. • Using a ruler, open your compasses to 9 cm. • Draw an arc on the tangent line TZ. Label the intersection point R. Step 4: Measure and state PR. • 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 cm. Since PT is a diameter, PT = 2 × OT = 2 × 4 cm = 8 cm. The tangent at T is perpendicular to the diameter PT at T. Therefore, PTR is a right-angled triangle with the right angle at T. We have PT = 8 cm and TR = 9 cm. Using the Pythagorean theorem: PR^2 = PT^2 + TR^2 PR^2 = (8 cm)^2 + (9 cm)^2 PR^2 = 64 cm^2 + 81 cm^2 PR^2 = 145 cm^2 PR = sqrt(145) cm As a decimal, PR ≈ 12.04 cm (to 2 decimal places). The length of PR is sqrt(145) cm ≈ 12.0 cm That's 2 down. 3 left today — send the next one.