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
Here's how to solve the problem: a) Calculate the length of the chord PQ. Step 1: Identify the right-angled triangle. The radius of the circle (r), the distance from the center to the midpoint of the chord (d), and half the length of the chord ((L)/(2)) form a right-angled triangle. Given: r = 10 cm and d = 6 cm. Step 2: Apply the Pythagorean theorem. r^2 = d^2 + ((L)/(2))^2 (10 cm)^2 = (6 cm)^2 + ((L)/(2))^2 100 cm^2 = 36 cm^2 + ((L)/(2))^2 Step 3: Solve for half the length of the chord. ((L)/(2))^2 = 100 cm^2 - 36 cm^2 ((L)/(2))^2 = 64 cm^2 (L)/(2) = sqrt(64 cm)^2 (L)/(2) = 8 cm Step 4: Calculate the full length of the chord PQ. L = 2 × 8 cm L = 16 cm The length of the chord PQ is 16 cm. b) Calculate the angle chord PQ subtends at the centre of the circle. Step 1: Consider the right-angled triangle formed by the center O, the midpoint M of the chord, and one endpoint P of the chord. Let be the angle subtended by the chord PQ at the center O (i.e., POQ). The line segment from O to M bisects this angle, so POM = ()/(2). In OMP: Hypotenuse OP = r = 10 cm Adjacent side OM = d = 6 cm Step 2: Use trigonometry to find half the central angle. We can use the cosine function: (()/(2)) = adjacenthypotenuse = (OM)/(OP) (()/(2)) = 6 cm10 cm (()/(2)) = 0.6 Step 3: Calculate the value of ()/(2). ()/(2) = (0.6) ()/(2) ≈ 53.13^ Step 4: Calculate the full central angle . = 2 × 53.13^ ≈ 106.26^ The angle chord PQ subtends at the center is 106.26^ (to two decimal places). 3 done, 2 left today. You're making progress.