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
You're on a roll — Step 1: Identify properties from the diagram. The figure shows a cyclic quadrilateral ABCD inscribed in a circle with center O. • AB is a diameter because it passes through the center O. • BCD = 110^. • ADC = t. • Chord CD = Chord CB (indicated by the double hash marks). Step 2: Use the property of angles in a semicircle. Since AB is a diameter, the angle subtended by the diameter at any point on the circumference is 90^. Therefore, ACB = 90^. Step 3: Calculate ACD. We know BCD = BCA + ACD. Substitute the known values: 110^ = 90^ + ACD ACD = 110^ - 90^ = 20^ Step 4: Use the property of equal chords. Given that chord CD = chord CB. Equal chords subtend equal arcs. Therefore, arc CD = arc CB. Angles subtended by equal arcs at the circumference are equal. So, CAD = CAB. Step 5: Use the property of opposite angles in a cyclic quadrilateral. In a cyclic quadrilateral, opposite angles sum to 180^. DAB + BCD = 180^ Substitute the given value for BCD: DAB + 110^ = 180^ DAB = 180^ - 110^ = 70^ Step 6: Calculate CAD. We know DAB = CAD + CAB. Since CAD = CAB (from Step 4), we can write: DAB = 2 × CAD 70^ = 2 × CAD CAD = (70^)/(2) = 35^ Step 7: Find t using the sum of angles in ADC. The sum of angles in a triangle is 180^. In ADC: ADC + CAD + ACD = 180^ Substitute the values for CAD and ACD: t + 35^ + 20^ = 180^ t + 55^ = 180^ t = 180^ - 55^ t = 125^ The value of t is 125^. This corresponds to option D. The final answer is D. 125^. Send me the next one 📸