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.
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Here are the solutions to your geometry problems. 1. Calculate the length of MC (x). Step 1: Find the length of AM. Since OM AB, M is the midpoint of AB. AM = (AB)/(2) = 8 cm2 = 4 cm Step 2: Find the length of OM using the Pythagorean theorem in OMA. OA is the radius, so OA = 5 cm. OA^2 = OM^2 + AM^2 (5 cm)^2 = OM^2 + (4 cm)^2 25 cm^2 = OM^2 + 16 cm^2 OM^2 = 25 cm^2 - 16 cm^2 OM^2 = 9 cm^2 OM = sqrt(9 cm)^2 = 3 cm Step 3: Calculate MC (x). OC is the radius, so OC = 5 cm. MC = OC - OM x = 5 cm - 3 cm x = 2 cm The length of MC is 2 cm. 2. Calculate the radius of the circle (x). Step 1: Identify the relationship between OM and AB. Since AM = MB = 15 units, M is the midpoint of chord AB. A 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. Step 2: Use the Pythagorean theorem in OMA. OA is the radius of the circle, denoted as x. OA^2 = OM^2 + AM^2 x^2 = (8 units)^2 + (15 units)^2 x^2 = 64 units^2 + 225 units^2 x^2 = 289 units^2 x = sqrt(289 units)^2 x = 17 units The radius of the circle is 17 units. 3. Calculate, giving reasons, the length of UA and TS. 3.1 Calculate UA. Step 1: Find the length of PQ. Since U is the center and UQ PR, Q is the midpoint of chord PR. (Reason: Line from center perpendicular to chord bisects the chord). PQ = (PR)/(2) = 8 cm2 = 4 cm Step 2: Find the length of UP (which is the radius, and UA is likely referring to the radius). In UQP, we have a right-angled triangle at Q. UP^2 = UQ^2 + PQ^2 UP^2 = (3 cm)^2 + (4 cm)^2 UP^2 = 9 cm^2 + 16 cm^2 UP^2 = 25 cm^2 UP = sqrt(25 cm)^2 UP = 5 cm Assuming UA refers to the radius of the circle (since U is the center and A is not defined as a specific point on the circumference), then UA = UP. The length of UA is 5 cm. 3.2 Calculate TS. Step 1: Find the length of the diameter TR. TR is the diameter, and U is the center. UP is the radius. TR = 2 × UP = 2 × 5 cm = 10 cm Step 2: Use the Pythagorean theorem in TSR. Since TR is the diameter, the angle subtended by the diameter at any point on the circumference is 90^. (Reason: Angle in a semicircle is 90^). Therefore, TSR = 90^. TR^2 = TS^2 + SR^2 (10 cm)^2 = (x cm)^2 + ((x+2) cm)^2 100 = x^2 + (x^2 + 4x + 4) 100 = 2x^2 + 4x + 4 0 = 2x^2 + 4x + 4 - 100 0 = 2x^2 + 4x - 96 Step 3: Solve the quadratic equation for x. Divide the entire equation by 2: 0 = x^2 + 2x - 48 Factor the quadratic equation: We need two numbers that multiply to -48 and add to 2. These numbers are 8 and -6. (x+8)(x-6) = 0 So, x+8=0 or x-6=0. x = -8 or x = 6 Since length cannot be negative, we discard x = -8. Therefore, x = 6 cm. Step 4: Calculate TS. TS = x cm. TS = 6 cm The length of TS is 6 cm. That's 2 down. 3 left today — send the next one.