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 given problems: 2.1.1 Determine Step 1: Identify the relevant right-angled triangle. Consider BDC. It is a right-angled triangle with B = 90^. The side opposite to is BD = 10 cm. The side adjacent to is BC = 40 cm. Step 2: Use the tangent function to find . = oppositeadjacent = (BD)/(BC) = (10)/(40) = (1)/(4) Step 3: Calculate the value of . = ((1)/(4)) ≈ 14.04^ The value of is 14.04^. 2.1.2 Determine x Step 1: Refer to the problem statement and the diagram. The problem states that BD = 10 cm. The diagram labels the length of the side BD as x. Step 2: State the value of x. x = 10 cm The value of x is 10 cm. 2.1.3 Determine Step 1: Find the total angle at C in the larger right-angled triangle. Consider ABC. It is a right-angled triangle with B = 90^ and A = 30^. The sum of angles in a triangle is 180^. BCA = 180^ - B - A BCA = 180^ - 90^ - 30^ BCA = 60^ Step 2: Use the relationship between BCA, , and . From the diagram, BCA = + . We found ≈ 14.04^ in 2.1.1. = BCA - = 60^ - 14.036^ ≈ 45.96^ The value of is 45.96^. 2.1.4 Determine Step 1: Identify the relevant triangle and known angles. Consider ADC. We know A = 30^ and ACD = ≈ 45.96^. The angle is ADC. Step 2: Use the sum of angles in ADC. The sum of angles in a triangle is 180^. A + ACD + ADC = 180^ 30^ + + = 180^ = 180^ - 30^ - = 150^ - 45.96^ ≈ 104.04^ The value of is 104.04^. 2.1.5 Determine AC Step 1: Identify the relevant right-angled triangle and known values. Consider ABC. It is a right-angled triangle with B = 90^. We know A = 30^ and the side opposite to it, BC = 40 cm. AC is the hypotenuse. Step 2: Use the sine function to find AC. A = oppositehypotenuse = (BC)/(AC) 30^ = (40)/(AC) Step 3: Solve for AC. AC = (40)/( 30^) AC = (40)/(0.5) AC = 80 cm The length of AC is 80 cm. 2.1.6 Determine AD Step 1: Find the length of the entire side AB. Consider ABC. We know A = 30^, B = 90^, and BC = 40 cm. We can use the tangent function. A = oppositeadjacent = (BC)/(AB) 30^ = (40)/(AB) Step 2: Solve for AB. AB = (40)/( 30^) AB = (40)/(1/sqrt(3)) AB = 40sqrt(3) cm Step 3: Calculate AD. From the diagram, D is a point on AB, so AD = AB - BD. We know BD = x = 10 cm from 2.1.2. AD = 40sqrt(3) - 10 AD ≈ 40(1.73205) - 10 AD ≈ 69.282 - 10 AD ≈ 59.28 cm The length of AD is (40sqrt(3) - 10) cm or approximately 59.28 cm. 3 done, 2 left today. You're making progress.