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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You're on a roll — here is the solution for Question 7: Question 7: In the diagram below, QRT is a straight line. If angle PTR = 90°, angle PRT = 60° and angle PQR = 30° and (PQ) = 6sqrt(3) cm. Calculate (RT). Given: QRT is a straight line. PTR = 90^ PRT = 60^ PQR = 30^ PQ = 6sqrt(3) cm Step 1: Find PRQ in PQR. Since QRT is a straight line, PRQ and PRT are supplementary angles. PRQ = 180^ - PRT PRQ = 180^ - 60^ PRQ = 120^ Step 2: Find QPR in PQR. The sum of angles in PQR is 180^. QPR = 180^ - PQR - PRQ QPR = 180^ - 30^ - 120^ QPR = 30^ Step 3: Determine the length of PR using the Sine Rule in PQR. Since PQR = QPR = 30^, PQR is an isosceles triangle, meaning PR = QR. Using the Sine Rule: (PR)/(( PQR)) = (PQ)/(( PRQ)) (PR)/((30^)) = 6sqrt(3)(120^) We know (30^) = (1)/(2) and (120^) = (180^ - 60^) = (60^) = sqrt(3)2. PR = 6sqrt(3) × (30^)(120^) PR = 6sqrt(3) × (1)/(2)sqrt(3)2 PR = 3sqrt(3)sqrt(3)2 PR = 3sqrt(3) × (2)/(sqrt(3)) PR = 6 cm Step 4: Calculate RT using trigonometry in the right-angled PRT. In PRT, we have PTR = 90^, PRT = 60^, and the hypotenuse PR = 6 cm. We want to find the adjacent side RT. ( PRT) = (RT)/(PR) (60^) = (RT)/(6) We know (60^) = (1)/(2). (1)/(2) = (RT)/(6) RT = 6 × (1)/(2) RT = 3 cm Drop the next question!