(a) In the diagram, TU is tangent to the circle. < RVU = 100° and < URS = 36°. Calculate the value of angle STU. (b) In triangle XYZ, |XY| = 5 cm, |YZ| = 8 ...

Answer Details

(a) Reading the diagram. \(R, V, S, U\) lie on the circle (\(R\) at top, \(V\) on the left, \(S\) on the right, \(U\) at the bottom). The line \(TU\) is the tangent that touches the circle at \(U\), and the secant through \(R\) and \(S\) is produced to meet the tangent at the external point \(T\). We are given \(\angle RVU = 100^\circ\) and \(\angle URS = 36^\circ\).

Step 1: Find arc \(SU\). \(\angle URS = 36^\circ\) is an inscribed angle at \(R\) standing on arc \(SU\) (the arc not containing \(R\)):

\[\text{arc } SU = 2\times 36^\circ = 72^\circ.\]

Step 2: Find arc \(RU\) on the far side. \(\angle RVU = 100^\circ\) is an inscribed angle at \(V\) standing on the arc \(RU\) that does not contain \(V\) (the arc going \(R\to S\to U\)):

\[\text{arc } RSU = \text{arc } RS + \text{arc } SU = 2\times 100^\circ = 200^\circ.\]

The remaining arc from \(R\) to \(U\) through \(V\) is therefore

\[\text{arc } RVU = 360^\circ - 200^\circ = 160^\circ.\]

Step 3: Apply the tangent-secant angle rule at \(T\). The angle between a tangent and a secant drawn from an external point equals half the difference of the two intercepted arcs. Here the far arc (between tangent point \(U\) and far point \(R\), through \(V\)) is \(160^\circ\) and the near arc (between \(U\) and near point \(S\)) is \(72^\circ\):

\[\angle STU = \tfrac{1}{2}\left(\text{arc } RVU - \text{arc } SU\right) = \tfrac{1}{2}(160^\circ - 72^\circ) = \tfrac{1}{2}(88^\circ) = 44^\circ.\]

Answer (a): \(\angle STU = 44^\circ\).

(b) Triangle \(XYZ\): \(|XY| = 5\), \(|YZ| = 8\), \(|XZ| = 6\text{ cm}\), with \(P\) on \(XY\), \(|XP| = 2\text{ cm}\), and \(PQ \parallel YZ\) with \(Q\) on \(XZ\).

Since \(PQ \parallel YZ\), triangle \(XPQ\) is similar to triangle \(XYZ\), so corresponding sides are proportional:

\[\frac{XP}{XY} = \frac{XQ}{XZ} \;\Rightarrow\; \frac{2}{5} = \frac{XQ}{6}.\]

\[XQ = \frac{2}{5}\times 6 = 2.4\text{ cm}.\]

\[|QZ| = |XZ| - |XQ| = 6 - 2.4 = 3.6\text{ cm}.\]

Answer (b): \(|QZ| = 3.6\text{ cm}\).