5. At one end \(A\) of a diameter \(AB\) of a circle of radius 5 cm, a tangent \(XAY\) is drawn. The length of the chord \(CD\) parallel to \(XY\) and at a distance 8 cm from \(A\) is
4 cm
5 cm
6 cm
8 cm
Step 1: Draw a circle with centre \(O\) and radius \(5\,\text{cm}\). Mark diameter \(AB\). At point \(A\), draw the tangent \(XAY\).
Step 2: We are told that a chord \(CD\) is drawn parallel to the tangent \(XY\). The perpendicular distance of this chord from point \(A\) is \(8\,\text{cm}\).
Step 3: To calculate chord length, we need the perpendicular distance of the chord from the centre \(O\). The total distance from \(O\) to point \(A\) is the radius = \(5\,\text{cm}\). The chord is \(8\,\text{cm}\) away from \(A\). Therefore, distance from \(O\) to the chord = \(|OA - 8| = |5 - 8| = 3\,\text{cm}|.
Step 4: Formula for chord length: \[ \text{Chord length} = 2 \sqrt{r^2 - d^2} \] where \(r\) is radius and \(d\) is distance from centre to the chord.
Step 5: Substitute values: \(r = 5\,\text{cm}, \; d = 3\,\text{cm}\) \[ \text{Chord length} = 2 \sqrt{5^2 - 3^2} = 2 \sqrt{25 - 9} = 2 \sqrt{16} = 2 \times 4 = 8\,\text{cm}. \]
Final Answer: The chord length = 8 cm. (Option D)