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 right-angled triangle and known lengths. In the given diagram, O is the center of the circle, and AB is a chord. M is the midpoint of AB. The line segment OM connects the center to the midpoint of the chord. A property of circles states that the line from the center to the midpoint of a chord is perpendicular to the chord. Therefore, OMA is a right-angled triangle with the right angle at M. We are given: • Radius OA = 13 units (hypotenuse of OMA) • Distance from center to chord OM = 5 units (one leg of OMA) We need to find the length of AM (the other leg of OMA). Step 2: Apply the Pythagorean theorem. For the right-angled triangle OMA: OA^2 = OM^2 + AM^2 Substitute the known values: 13^2 = 5^2 + AM^2 Step 3: Solve for AM. 169 = 25 + AM^2 AM^2 = 169 - 25 AM^2 = 144 AM = sqrt(144) AM = 12 units Step 4: Calculate the length of chord AB. Since M is the midpoint of chord AB, the length of AB is twice the length of AM. AB = 2 × AM AB = 2 × 12 AB = 24 units Reasons: • The line from the center of a circle to the midpoint of a chord is perpendicular to the chord. Thus, OMA is a right-angled triangle at M. • The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (radius OA) is equal to the sum of the squares of the other two sides (OM and AM). • M is the midpoint of AB, so AB = 2 × AM. The length of chord AB is 24 units. Send me the next one 📸