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
Welcome back mrinal — missed you this week. Step 1: Identify given information and properties. Given: A circle with center O. XY and X'Y' are two parallel tangents. AB is another tangent at point C, intersecting XY at A and X'Y' at B. P is the point of tangency for XY, and Q is the point of tangency for X'Y'. To prove: AOB = 90^. We use the property that tangents drawn from an external point to a circle are equal in length, and the line segment joining the center to the external point bisects the angle between the tangents. Step 2: Apply tangent properties for point A. From external point A, AP and AC are tangents to the circle. Therefore, AP = AC. Consider OPA and OAC: • OP = OC (radii of the same circle) • OA = OA (common side) • AP = AC (tangents from external point A) By SSS congruence criterion, OPA OAC. Thus, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), POA = COA. Let COA = . Then POA = . Step 3: Apply tangent properties for point B. From external point B, BQ and BC are tangents to the circle. Therefore, BQ = BC. Consider OQB and OCB: • OQ = OC (radii of the same circle) • OB = OB (common side) • BQ = BC (tangents from external point B) By SSS congruence criterion, OQB OCB. Thus, by CPCTC, QOB = COB. Let COB = . Then QOB = . Step 4: Use the property of parallel tangents. Since XY and X'Y' are parallel tangents, the line segment PQ passing through the center O is a diameter. This means P, O, and Q are collinear. Therefore, the angle POQ is a straight angle, i.e., POQ = 180^. We can express POQ as the sum of the angles around O: POQ = POA + COA + COB + QOB Substitute the values from Step 2 and Step 3: 180^ = + + + 180^ = 2 + 2 180^ = 2( + ) Divide by 2: + = 90^ Step 5: Conclude the proof. The angle AOB is the sum of COA and COB. AOB = COA + COB Substitute the values from Step 2 and Step 3: AOB = + From Step 4, we found that + = 90^. Therefore, AOB = 90^ Hence, it is proven that AOB = 90^. Send me the next one 📸