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
Step 1: Identify similar triangles using QR WT. Given that QR WT. Consider QRS and WTS. Since WT QR, we have: S is common to both triangles. STW = SRQ (corresponding angles). Therefore, WTS QRS by AA similarity. From the similarity, the ratio of corresponding sides is equal: (WS)/(QS) = (ST)/(SR) We are given QW = 12 mm and WS = 11 mm. The length of QS is QW + WS = 12 + 11 = 23 mm. Substitute the known values into the ratio: (11)/(23) = (ST)/(SR) (*) Step 2: Identify similar triangles using QT WU. Given that QT WU. Consider QST and WSU. Since WU QT, we have: S is common to both triangles. SWU = SQT (corresponding angles). Therefore, WSU QST by AA similarity. From the similarity, the ratio of corresponding sides is equal: (SW)/(SQ) = (SU)/(ST) We know SW = 11 mm and SQ = 23 mm. Substitute these values into the ratio: (11)/(23) = (SU)/(ST) Step 3: Calculate ST. From the ratio in Step 2, we can express SU in terms of ST: SU = (11)/(23) ST We are given TU = 6 mm. From the diagram, points S, U, T are collinear on RS in that order, so ST = SU + UT. Substitute the expression for SU and the value of TU: ST = (11)/(23) ST + 6 Subtract (11)/(23) ST from both sides: ST - (11)/(23) ST = 6 (23 ST - 11 ST)/(23) = 6 (12 ST)/(23) = 6 Multiply both sides by 23 and divide by 12: 12 ST = 6 × 23 12 ST = 138 ST = (138)/(12) ST = 11.5 mm Step 4: Calculate RS. Substitute the value of ST = 11.5 mm into equation (*) from Step 1: (11)/(23) = (11.5)/(SR) Multiply both sides by 23 × SR: 11 × SR = 23 × 11.5 11 SR = 264.5 Divide by 11: SR = (264.5)/(11) SR = 24.04545... mm Step 5: Round to the nearest whole number. Rounding 24.04545... to the nearest whole number gives 24. RS ≈ 24 mm The final answer is 24 mm. Send me the next one 📸