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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Here are the solutions for the problems you've sent: 1.1 In the right-angled triangles below, calculate the length of the sides that have not been given. Leave your answers in surd form where applicable. a) Step 1: Apply the Pythagorean theorem. For a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse. Given sides are 15 and 20 (hypotenuse). Let the unknown side be y. y^2 + 15^2 = 20^2 Step 2: Calculate the squares and solve for y. y^2 + 225 = 400 y^2 = 400 - 225 y^2 = 175 Step 3: Find the square root and simplify the surd. y = sqrt(175) y = sqrt(25 × 7) y = 5sqrt(7) The length of the side is 5sqrt(7). b) Step 1: Apply the Pythagorean theorem. Given sides are 24 and 25 (hypotenuse). Let the unknown side be x. x^2 + 24^2 = 25^2 Step 2: Calculate the squares and solve for x. x^2 + 576 = 625 x^2 = 625 - 576 x^2 = 49 Step 3: Find the square root. x = sqrt(49) x = 7 The length of the side is 7. 1.2 A ladder with a length of 10m is placed at an angle against a wall. The bottom of the ladder is 2 meters away from the wall. How far up the wall will the ladder reach? Round off to two decimal places. Step 1: Identify the sides of the right-angled triangle. The ladder forms the hypotenuse (c = 10 m). The distance from the wall is one leg (a = 2 m). The height the ladder reaches up the wall is the other leg (h). Apply the Pythagorean theorem: a^2 + h^2 = c^2. 2^2 + h^2 = 10^2 Step 2: Calculate the squares and solve for h. 4 + h^2 = 100 h^2 = 100 - 4 h^2 = 96 Step 3: Find the square root and round to two decimal places. h = sqrt(96) h ≈ 9.79795 h ≈ 9.80 m The ladder will reach approximately 9.80 m up the wall. 1.3 RQ is a vertical pole. The foot of the pole, Q, is on the same horizontal plane as P and S. The pole is anchored with cables RS and RP. The angle of depression from the top of the pole to the point S is 47^. RS is 20m and QS is 17m. RSQ = . PQ = QS. a) Calculate the size of . Step 1: Understand the angle of depression. The angle of depression from R to S is the angle between the horizontal line from R (let's call it RT) and the line of sight RS. So, TRS = 47^. Step 2: Use properties of parallel lines. Since RQ is a vertical pole and QS is on a horizontal plane, RQ is perpendicular to QS. The horizontal line RT is parallel to QS. Therefore, RSQ and TRS are alternate interior angles. RSQ = TRS = 47^ So, = 47^. b) Calculate the length of RQ. Step 1: Identify the right-angled triangle. RQS is a right-angled triangle at Q (since RQ is vertical and QS is horizontal). We are given the hypotenuse RS = 20 m and leg QS = 17 m. We need to find leg RQ. Step 2: Apply the Pythagorean theorem. RQ^2 + QS^2 = RS^2 RQ^2 + 17^2 = 20^2 Step 3: Calculate the squares and solve for RQ. RQ^2 + 289 = 400 RQ^2 = 400 - 289 RQ^2 = 111 RQ = sqrt(111) m The length of RQ is sqrt(111) m. c) Calculate the length of RP. Step 1: Identify the right-angled triangle and its sides. We are given PQ = QS. Since QS = 17 m, then PQ = 17 m. RQP is a right-angled triangle at Q. We have leg RQ = sqrt(111) m (from part b) and leg PQ = 17 m. We need to find the hypotenuse RP. Step 2: Apply the Pythagorean theorem. RP^2 = RQ^2 + PQ^2 RP^2 = (sqrt(111))^2 + 17^2 Step 3: Calculate the squares and solve for RP. RP^2 = 111 + 289 RP^2 = 400 RP = sqrt(400) RP = 20 m The length of RP is 20 m. That's 2 down. 3 left today — send the next one.