O-Level Proofs in plane geometry

Angle PZT = 90° is the angle subtended by the chord PT at the point Z on the circle. By the converse of the angle-in-a-semicircle theorem, a chord that subtends a right angle at a point on the circle must be a diameter. So PT is a diameter, and the centre of the circle is the midpoint of any diameter, hence the midpoint of PT.

Method: write h in terms of r using similar triangles from a vertical cross-section of the cone, then substitute into the cylinder volume formula. Step 1 - set up the cross-section. Slice the cone vertically through its axis. The cross-section is a triangle of height 45 cm with half-base 20 cm (the base radius). The cylinder appears as a rectangle of width 2r sitting on the base, reaching up to height h. Step 2 - identify the similar triangles. The top rim of the cylinder just touches the sloping side. Above that rim sits a smaller triangle, cut off by a line parallel to the base, so it is similar to the whole cone. Small triangle: height 45 − h, half-base r. Whole cone: height 45, half-base 20. Step 3 - equate the ratios of corresponding sides. (45 − h)/45 = r/20 Step 4 - make h the subject. Multiply both sides by 45: 45 − h = 45r/20 45/20 simplifies to 9/4, so 45 − h = (9/4)r h = 45 − (9/4)r Step 5 - substitute into the volume of a cylinder. V = πr²h V = πr²[45 − (9/4)r] Expand the bracket: V = 45πr² − (9/4)πr³ which is the required result.