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 problems 27 and 28. Problem 27: a) Using a ruler, a pair of compasses and a sharp pencil construct an equilateral triangle in a circle of radius 4cm. Step 1: Draw a circle. Using a ruler, measure 4 cm. Set your compass to this radius. Draw a circle with any center point O. Step 2: Mark a point and an arc. Mark any point A on the circumference of the circle. With A as the center and the same radius (4 cm), draw an arc that intersects the circle at two points. Let's call one of these intersection points B. Step 3: Draw another arc. With B as the center and the same radius (4 cm), draw another arc that intersects the circle at a new point. Let's call this point C. Step 4: Connect the vertices. Connect points A, C, and the point diametrically opposite to B (which would be the third vertex if you continued marking points around the circle, or simply connect A, C, and the point that is two compass-widths away from A in the other direction). A more direct method: 1. Draw a circle with center O and radius 4 cm. 2. Draw a diameter through O, marking points P and Q on the circumference. 3. With P as the center and radius 4 cm, draw an arc that intersects the circle at two points, say R and S. 4. Connect points Q, R, and S. Triangle QRS is an equilateral triangle inscribed in the circle. b) What is the length of the triangle? Step 1: Recall the formula for the side length of an equilateral triangle inscribed in a circle. For an equilateral triangle inscribed in a circle with radius R, the side length s is given by the formula: s = Rsqrt(3) Step 2: Substitute the given radius. Given radius R = 4 cm. s = 4sqrt(3) cm Step 3: Calculate the approximate numerical value (optional, but good for understanding). Using sqrt(3) ≈ 1.732: s ≈ 4 × 1.732 cm s ≈ 6.928 cm The length of the side of the equilateral triangle is 4sqrt(3) cm. Problem 28: Study the figure below and answer the questions that follow. The figure is a trapezium with parallel sides of 14 m and 26 m, and a height of 8 m. a) Find the area of the figure. Step 1: Identify the lengths of the parallel sides and the height. The lengths of the parallel sides are a = 14 m and b = 26 m. The height of the trapezium is h = 8 m. Step 2: Apply the formula for the area of a trapezium. The area of a trapezium is given by the formula: Area = (1)/(2)(a+b)h Step 3: Substitute the values and calculate the area. Area = (1)/(2)(14 m + 26 m) × 8 m Area = (1)/(2)(40 m) × 8 m Area = 20 m × 8 m Area = 160 m^2 The area of the figure is 160 m^2. Drop the next question!