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 to the problems: 11. Define the term triangle. A triangle is a polygon with three straight sides and three vertices. It is a closed two-dimensional shape whose interior angles sum to 180^. 12. Classify the following triangles with angles: a) 60^, 60^, 60^: Equilateral triangle (all angles are equal, implying all sides are equal). b) 70^, 40^, 70^: Isosceles triangle (two angles are equal, implying two sides are equal). c) 40^, 72^, 68^: Scalene triangle (all angles are different, implying all sides are different). 13. Calculate both the perimeter and area for the following: a) Step 1: Calculate the perimeter. The sides of the triangle are 9 cm, 12 cm, and 15 cm. Perimeter = 9 cm + 12 cm + 15 cm Perimeter = 36 cm Step 2: Calculate the area. The base is 15 cm and the height is 6 cm. Area = (1)/(2) × base × height Area = (1)/(2) × 15 cm × 6 cm Area = (1)/(2) × 90 cm^2 Area = 45 cm^2 b) Step 1: Calculate the area. The base is 7 cm and the height is 4 cm. Area = (1)/(2) × base × height Area = (1)/(2) × 7 cm × 4 cm Area = (1)/(2) × 28 cm^2 Area = 14 cm^2 Note: The perimeter cannot be calculated without the length of the third side. The given side 2.5 cm is one of the non-base sides. c) The given side lengths are 3 cm, 3 cm, and 8 cm. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Here, 3 cm + 3 cm = 6 cm. Since 6 cm < 8 cm, a triangle cannot be formed with these dimensions. 14. In the triangle below, angles at A and B are 55^ and 115^ respectively. Find the value of x at C. Step 1: Find the interior angle at B. The exterior angle at B is 115^. The interior angle at B and the exterior angle at B are supplementary (they add up to 180^). Interior Angle at B = 180^ - 115^ Interior Angle at B = 65^ Step 2: Use the sum of angles in a triangle. The sum of the interior angles of a triangle is 180^. Angle A + Interior Angle B + Angle C = 180^ 55^ + 65^ + x = 180^ 120^ + x = 180^ Step 3: Solve for x. x = 180^ - 120^ x = 60^