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.
ScanSolve AI Answer
Welcome back maame — missed you this week. Here are the solutions to your questions: 5. a) Step 1: Recall the formula for the area of an equilateral triangle. The area A of an equilateral triangle with side length s is given by: A = sqrt(3)4 s^2 Step 2: Substitute the given area and solve for the side length s. We are given A = 53sqrt(3) cm^2. 53sqrt(3) = sqrt(3)4 s^2 Divide both sides by sqrt(3): 53 = (1)/(4) s^2 Multiply both sides by 4: s^2 = 53 × 4 s^2 = 212 Take the square root of both sides: s = sqrt(212) Simplify the radical: s = sqrt(4 × 53) s = 2sqrt(53) cm Step 3: Calculate the perimeter. The perimeter P of an equilateral triangle is 3s. P = 3 × (2sqrt(53)) P = 6sqrt(53) cm The perimeter of the equilateral triangle is 6sqrt(53) cm. 5. b) Step 1: Find the length of the hypotenuse AB using the Pythagorean theorem in ABC. Given |AC| = 6 cm and |BC| = 8 cm. AB^2 = AC^2 + BC^2 AB^2 = 6^2 + 8^2 AB^2 = 36 + 64 AB^2 = 100 AB = sqrt(100) AB = 10 cm Step 2: Use the property of similar triangles to find x. In a right-angled triangle, the altitude to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other. Thus, ADC ACB. From the similarity, the ratio of corresponding sides is equal: (AD)/(AC) = (AC)/(AB) Substitute the known values: (x)/(6) = (6)/(10) Step 3: Solve for x. Multiply both sides by 6: x = (6 × 6)/(10) x = (36)/(10) x = 3.6 cm The value of x is 3.6 cm. Send me the next one 📸