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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You're on a roll — here are the solutions to the problems. 18. The figure below shows a triangle inside a circle. AB = 8 cm, BC = 10 cm and AC = 13 cm. a) Calculate the area of triangle ABC. Step 1: Calculate the semi-perimeter (S) of the triangle. The side lengths are a = BC = 10 cm, b = AC = 13 cm, and c = AB = 8 cm. S = (1)/(2)(a+b+c) = (1)/(2)(10 + 13 + 8) = (1)/(2)(31) = 15.5 cm Step 2: Use Heron's formula to calculate the area of ABC. Area = sqrt(S(S-a)(S-b)(S-c)) Area = sqrt(15.5(15.5-10)(15.5-13)(15.5-8)) Area = sqrt(15.5(5.5)(2.5)(7.5)) Area = sqrt(1598.4375) Area ≈ 39.98046 cm^2 Rounding to two decimal places: The area of triangle ABC is 39.98 cm^2. b) Angle BAC Step 1: Use the area formula involving sine to find BAC. Let = BAC. The sides adjacent to BAC are AB=c=8 cm and AC=b=13 cm. Area = (1)/(2)bc 39.98046 = (1)/(2) × 13 × 8 × 39.98046 = 52 Step 2: Solve for and then . = (39.98046)/(52) ≈ 0.768855 = (0.768855) ≈ 50.24^ Rounding to two decimal places: Angle BAC is 50.24^. c) The radius of the circle. Step 1: Use the formula for the circumradius (R) of a