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's the solution for Question 4: Given the sides of the triangle are a = 13 cm, b = 14 cm, and c = 15 cm. a) Calculate the semi-perimeter s. Step 1: Sum the lengths of the sides. a + b + c = 13 cm + 14 cm + 15 cm = 42 cm Step 2: Divide the sum by 2 to find the semi-perimeter. s = (a + b + c)/(2) = 42 cm2 = 21 cm The semi-perimeter s is 21 cm. b) Calculate (s - a), (s - b), and (s - c). Step 1: Calculate (s - a). s - a = 21 cm - 13 cm = 8 cm Step 2: Calculate (s - b). s - b = 21 cm - 14 cm = 7 cm Step 3: Calculate (s - c). s - c = 21 cm - 15 cm = 6 cm The values are (s - a) = 8 cm, (s - b) = 7 cm, and (s - c) = 6 cm. c) Apply Heron's Formula to find the area. Step 1: Substitute the values into Heron's Formula. Area = sqrt(s(s-a)(s-b)(s-c)) Area = sqrt(21 cm) × 8 cm × 7 cm × 6 cm Step 2: Multiply the terms under the square root. Area = sqrt(7056 cm)^4 Step 3: Calculate the square root. Area = 84 cm^2 The area of the triangle is 84.00 cm^2 (correct to 2 decimal places). d) Find the new area if all three sides are doubled. Step 1: Determine the linear scale factor. The sides are doubled, so the linear scale factor k = 2. Step 2: Determine the area scale factor. The area scale factor is k^2. Area scale factor = (2)^2 = 4 Step 3: Multiply the original area by the area scale factor to find the new area. New Area = Original Area × Area scale factor New Area = 84 cm^2 × 4 New Area = 336 cm^2 The new area is 336 cm^2. What's next?