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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Question 6 6.1 Step 1: Identify properties of PTU. Given that PQRS is a square, P = 90^. It is also given that PT = PU. Therefore, PTU is a right-angled isosceles triangle. Step 2: Calculate the base angles of PTU. The sum of angles in a triangle is 180^. PTU = PUT = (180^ - 90^)/(2) = 45^ Step 3: Identify properties of the diagonal QS. In a square PQRS, the diagonal QS bisects PQR and PSR. Thus, PQS = 45^. Step 4: Compare angles to prove parallelism. We have PTU = 45^ and PQS = 45^. Since PTU = PQS, and these are corresponding angles with transversal PQ, we can conclude that TU QS. 6.2 Step 1: Identify the dimensions of PQS. Given PQ = 10 units. Since PQRS is a square, PS = PQ = 10 units. PQS is a right-angled triangle at P. Step 2: Calculate the area of PQS. The area of a right-angled triangle is (1)/(2) × base × height. Area of PQS = (1)/(2) × PQ × PS Area of PQS = (1)/(2) × 10 units × 10 units Area of PQS = (1)/(2) × 100 units^2 Area of PQS = 50 units^2 6.3 Step 1: Prove that QRS is a right-angled isosceles triangle. Since PQRS is a square, all its sides are equal in length (QR = RS) and all its interior angles are 90^ ( R = 90^). Therefore, QRS has two equal sides (QR and RS) and one right angle ( R), which means it is a right-angled isosceles triangle. Step 2: Determine the length of QS. Using the Pythagorean theorem in QRS: QS^2 = QR^2 + RS^2 Since QR = RS = PQ = 10 units (from the properties of a square and the given information in 6.2): QS^2 = (10 units)^2 + (10 units)^2 QS^2 = 100 units^2 + 100 units^2 QS^2 = 200 units^2 QS = sqrt(200) units QS = sqrt(100 × 2) units QS = 10sqrt(2) units That's 2 down. 3 left today — send the next one.