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
Here are the steps to construct PQR and calculate its angles. Given side lengths: PQ = 8 cm QR = 11 cm PR = 10 cm Step 1: Check if the triangle can be constructed. We apply the Triangle Inequality Theorem: the sum of any two sides must be greater than the third side. PQ + QR > PR 8 + 11 = 19 cm. Since 19 > 10, this condition is met. PQ + PR > QR 8 + 10 = 18 cm. Since 18 > 11, this condition is met. QR + PR > PQ 11 + 10 = 21 cm. Since 21 > 8, this condition is met. Since all conditions are met, the triangle can be constructed. Step 2: Describe the construction steps. 1. Draw a line segment PQ of length 8 cm. 2. With P as the center, use a compass to draw an arc with a radius of 10 cm (the length of PR). 3. With Q as the center, use a compass to draw an arc with a radius of 11 cm (the length of QR). 4. The point where these two arcs intersect is point R. 5. Draw line segments PR and QR to complete PQR. Step 3: Calculate the angles using the Law of Cosines. Let p = QR = 11 cm, q = PR = 10 cm, and r = PQ = 8 cm. To find P: P = (q^2 + r^2 - p^2)/(2qr) P = (10^2 + 8^2 - 11^2)/(2 · 10 · 8) P = (100 + 64 - 121)/(160) P = (43)/(160) P = ((43)/(160)) P ≈ 74.38^ To find Q: Q = (p^2 + r^2 - q^2)/(2pr) Q = (11^2 + 8^2 - 10^2)/(2 · 11 · 8) Q = (121 + 64 - 100)/(176) Q = (85)/(176) Q = ((85)/(176)) Q ≈ 61.05^ To find R: R = (p^2 + q^2 - r^2)/(2pq) R = (11^2 + 10^2 - 8^2)/(2 · 11 · 10) R = (121 + 100 - 64)/(220) R = (157)/(220) R = ((157)/(220)) R ≈ 44.57^ The sum of the angles is 74.38^ + 61.05^ + 44.57^ = 180.00^. That's 2 down. 3 left today — send the next one.