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
To calculate |RT|, we will use trigonometric ratios in the given triangles. Step 1: Identify the angles in PQT. We are given that PTR = 90^, which means PQT is a right-angled triangle at T. We are given PQR = 30^. The sum of angles in PQT is 180^. So, QPT = 180^ - PTQ - PQT = 180^ - 90^ - 30^ = 60^. Step 2: Use trigonometry in PQT to find the length of |PT|. We know |PQ| = 6sqrt(3) cm and PQT = 30^. In PQT, |PT| is the side opposite to PQT, and |PQ| is the hypotenuse. We use the sine function: ( PQT) = (|PT|)/(|PQ|) (30^) = (|PT|)/(6sqrt(3) cm) We know that (30^) = (1)/(2). (1)/(2) = (|PT|)/(6sqrt(3) cm) |PT| = (1)/(2) × 6sqrt(3) cm |PT| = 3sqrt(3) cm Step 3: Use trigonometry in PRT to find the length of |RT|. We know that PRT is a right-angled triangle at T. We are given PRT = 60^. We found |PT| = 3sqrt(3) cm. In PRT, |PT| is the side opposite to PRT, and |RT| is the side adjacent to PRT. We use the tangent function: ( PRT) = (|PT|)/(|RT|) (60^) = 3sqrt(3) cm|RT| We know that (60^) = sqrt(3). sqrt(3) = 3sqrt(3) cm|RT| Now, solve for |RT|: |RT| = 3sqrt(3) cmsqrt(3) |RT| = 3 cm Comparing this with the given options, the correct option is C. The final answer is C. 3cm. That's 2 down. 3 left today — send the next one.