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, based on the interpretation of the given side lengths for question 1. Given side lengths for question 1: PQ = 6 cm BC = 6 cm AC = 10 cm Step 1: Interpret the side lengths. The question asks to construct PQR but provides side lengths PQ, BC, and AC. Assuming BC refers to QR and AC refers to PR for PQR, the side lengths are: PQ = 6 cm QR = 6 cm PR = 10 cm Step 2: 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 6 + 6 = 12 cm. Since 12 > 10, this condition is met. PQ + PR > QR 6 + 10 = 16 cm. Since 16 > 6, this condition is met. QR + PR > PQ 6 + 10 = 16 cm. Since 16 > 6, this condition is met. Since all conditions are met, the triangle can be constructed. Step 3: Describe the construction steps. 1. Draw a line segment PR of length 10 cm. 2. With P as the center, use a compass to draw an arc with a radius of 6 cm (the length of PQ). 3. With R as the center, use a compass to draw an arc with a radius of 6 cm (the length of QR). 4. The point where these two arcs intersect is point Q. 5. Draw line segments PQ and QR to complete PQR. Step 4: Calculate the angles using the Law of Cosines. Let p = QR = 6 cm, q = PR = 10 cm, and r = PQ = 6 cm. To find P: P = (q^2 + r^2 - p^2)/(2qr) P = (10^2 + 6^2 - 6^2)/(2 · 10 · 6) P = (100 + 36 - 36)/(120) P = (100)/(120) P = (5)/(6) P = ((5)/(6)) P ≈ 33.56^ To find R: Since PQ = QR = 6 cm, PQR is an isosceles triangle, which means P = R. R ≈ 33.56^ To find Q: The sum of angles in a triangle is 180^. Q = 180^ - P - R Q = 180^ - 33.56^ - 33.56^ Q = 180^ - 67.12^ Q ≈ 112.88^ The sum of the angles is 33.56^ + 33.56^ + 112.88^ = 180.00^. Note: The diagram on the page shows an angle of 60^ at R, but this contradicts the side lengths given in question 1, so it is not used for this problem. 3 done, 2 left today. You're making progress.