Question

(c) the diagram below shows a vertical line pt.
the points b, c, d are all joined to t.
the points p, b, c, d form a rectangle.
the lines pb and pd are both perpendicular to pt.
pt = 8, dc = 12, bc = 10
all lengths are in metres.
calculate the length of the line joining c to t, showing

Explanation:

Step1: Find the length of \(PC\)

Since \(P, B, C, D\) form a rectangle, \(PC = BD\) (diagonals of a rectangle are equal). In a rectangle, if \(DC = 12\) and \(BC=10\), by the Pythagorean theorem \(BD=\sqrt{BC^{2}+DC^{2}}\).

$$BD=\sqrt{10^{2}+12^{2}}=\sqrt{100 + 144}=\sqrt{244}$$

Step2: Use the Pythagorean theorem in right - triangle \(PCT\)

Since \(PT = 8\) and \(PC=\sqrt{244}\), and \(\angle TPC = 90^{\circ}\) (because \(PB\) and \(PD\) are perpendicular to \(PT\) and \(P, B, C, D\) is a rectangle), by the Pythagorean theorem \(CT=\sqrt{PT^{2}+PC^{2}}\).
Substitute \(PT = 8\) and \(PC=\sqrt{244}\) into the formula:

$$CT=\sqrt{8^{2}+244}=\sqrt{64 + 244}=\sqrt{308}=\sqrt{4\times77}=2\sqrt{77}\approx 17.55$$

Answer:

\(CT\approx17.6\) meters (rounded to one decimal place)