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 your working.

Explanation:

Step1: Use the Pythagorean theorem

Since \(PB\) and \(PD\) are perpendicular to \(PT\) and \(P, B, C, D\) form a rectangle, in right - triangle \(TBC\), we know that \(TB^{2}=TP^{2}+PB^{2}\) (by Pythagoras in \(\triangle TPB\)), and \(PB = DC = 12\) (opposite sides of a rectangle), \(TP = 8\). First, find \(TB\):

$$TB=\sqrt{8^{2}+12^{2}}=\sqrt{64 + 144}=\sqrt{208}$$

In right - triangle \(TBC\), we know \(BC = 10\). Now, use the Pythagorean theorem in \(\triangle TBC\) to find \(TC\). The Pythagorean theorem states that for a right - triangle with sides \(a,b\) and hypotenuse \(c\), \(c=\sqrt{a^{2}+b^{2}}\). Here, if \(a = TB=\sqrt{208}\) and \(b = BC = 10\), then \(TC=\sqrt{TB^{2}+BC^{2}}\)
Substitute \(TB^{2}=208\) and \(BC^{2}=100\) into the formula:

$$TC=\sqrt{208+100}$$

Step2: Simplify the expression

$$TC=\sqrt{308}=\sqrt{4\times77}=2\sqrt{77}\approx 17.55$$

Answer:

\(TC\approx17.55\) meters