This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given information. A circle with center O has a radius r = 5 cm. A chord PQ has a length of 6 cm. i) Calculate the distance of the chord from the Centre O. Step 1: Draw a diagram and identify relevant geometric properties. Let M be the midpoint of the chord PQ. The line segment OM represents the perpendicular distance from the center O to the chord PQ. In a circle, the line from the center to the midpoint of a chord is perpendicular to the chord. Therefore, OMP is a right-angled triangle with the right angle at M. Step 2: Determine the lengths of the sides of OMP. The hypotenuse OP is the radius of the circle, so OP = 5 cm. The side PM is half the length of the chord PQ. PM = (PQ)/(2) = (6)/(2) = 3 cm The side OM is the distance we need to find. Step 3: Apply the Pythagorean theorem. In OMP: OP^2 = OM^2 + PM^2 Substitute the known values: 5^2 = OM^2 + 3^2 25 = OM^2 + 9 OM^2 = 25 - 9 OM^2 = 16 OM = sqrt(16) OM = 4 cm The distance of the chord from the center O is 4 cm. ii) Calculate Angle POQ. Step 1: Relate POQ to OMP. The angle POQ is the angle subtended by the chord PQ at the center O. Since OM bisects POQ, we have POQ = 2 × POM. Let POM = . Step 2: Use trigonometry in OMP to find . In the right-angled triangle OMP: We know OP = 5 cm (hypotenuse), PM = 3 cm (opposite to ), and OM = 4 cm (adjacent to ). We can use the sine function: () = oppositehypotenuse = (PM)/(OP) () = (3)/(5) = ((3)/(5)) Using a calculator: ≈ 36.86989^ Step 3: Calculate POQ. POQ = 2 × POQ = 2 × ((3)/(5)) POQ ≈ 2 × 36.86989^ POQ ≈ 73.73978^ Rounding to one decimal place: POQ ≈ 73.7^ The angle POQ is 73.7^.