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 to questions 4 to 8: 4. In ABC, m C = 42^, a = 19 and b = 26. Find the length of side c, to the nearest integer. Step 1: Use the Law of Cosines to find the length of side c. The formula is c^2 = a^2 + b^2 - 2ab (C). c^2 = 19^2 + 26^2 - 2(19)(26) (42^) Step 2: Calculate the squares and the product. c^2 = 361 + 676 - 988 (42^) c^2 = 1037 - 988 (42^) Step 3: Calculate the value of (42^) and substitute. (42^) ≈ 0.74314 c^2 = 1037 - 988(0.74314) c^2 = 1037 - 734.19872 c^2 = 302.80128 Step 4: Take the square root to find c and round to the nearest integer. c = sqrt(302.80128) c ≈ 17.401 Rounding to the nearest integer, c = 17. Answer: The length of side c is 17. 5. In a rhombus whose side measures 22 cm and the smaller angle is 55^, find the length of the larger diagonal, to the nearest tenth. Step 1: Determine the larger interior angle of the rhombus. The sum of consecutive angles in a rhombus is 180^. Larger angle = 180^ - 55^ = 125^. Step 2: Consider one of the right-angled triangles formed by the diagonals. The diagonals of a rhombus bisect each other at right angles and bisect the angles of the rhombus. The hypotenuse of this right triangle is the side of the rhombus, which is 22 cm. The angle opposite half of the larger diagonal is half of the larger angle, so (125^)/(2) = 62.5^. Step 3: Use the sine function to find half the length of the larger diagonal (d_L/2). (62.5^) = (d_L/2)/(22) d_L/2 = 22 (62.5^) d_L/2 = 22 × 0.88701 d_L/2 = 19.51422 Step 4: Calculate the full length of the larger diagonal and round to the nearest tenth. d_L = 2 × 19.51422 d_L = 39.02844 Rounding to the nearest tenth, d_L = 39.0 cm. Answer: The length of the larger diagonal is 39.0 cm. 6. Use Heron's formula to find the area of a triangle of lengths 7, 8 and 9. Give your answer in surd form. Step 1: Calculate the semi-perimeter (s). s = (a+b+c)/(2) = (7+8+9)/(2) = (24)/(2) = 12 Step 2: Apply Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)). Area = sqrt(12(12-7)(12-8)(12-9)) Area = sqrt(12(5)(4)(3)) Step 3: Multiply the terms under the square root. Area = sqrt(12 × 5 × 4 × 3) Area = sqrt(60 × 12) Area = sqrt(720) Step 4: Simplify the surd. Find the largest perfect square factor of 720. 720 = 144 × 5 Area = sqrt(144 × 5) Area = sqrt(144) × sqrt(5) Area = 12sqrt(5) Answer: The area of the triangle is 12sqrt(5). 7. What is the area of a triangle where every side is 5 units long correct to the nearest unit? Step 1: Recognize that a triangle with all sides equal is an equilateral triangle. The side length s = 5 units. Step 2: Use the formula for the area of an equilateral triangle: Area = sqrt(3)4 s^2. Area = sqrt(3)4 (5^2) Area = sqrt(3)4 (25) Area = 25sqrt(3)4 Step 3: Calculate the numerical value and round to the nearest unit. sqrt(3) ≈ 1.73205 Area = (25 × 1.73205)/(4) Area = (43.30125)/(4) Area = 10.8253125 Rounding to the nearest unit, the area is 11. Answer: The area of the triangle is 11 units^2. 8. Calculate the exact area of ABC. Step 1: Identify the given information from the diagram. Side AC = 5 units. Side BC = 6 units. Included angle C = 60^. Step 2: Use the formula for the area of a triangle given two sides and the included angle: Area = (1)/(2)ab (C). Area = (1)/(2) (AC)(BC) (C) Area = (1)/(2) (5)(6) (60^) Step 3: Substitute the exact value of (60^). (60^) = sqrt(3)2 Area = (1)/(2) (30) (sqrt(3)2) Step 4: Simplify the expression. Area = 15 (sqrt(3)2) Area = 15sqrt(3)2 Answer: The exact area of ABC is 15sqrt(3)2. That's 2 down. 3 left today — send the next one.