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
Step 1: Understand the properties of a rhombus. A rhombus has four equal sides. Its diagonals bisect each other at right angles. This means that the intersection of the diagonals forms four right-angled triangles, where the legs are half the lengths of the diagonals and the hypotenuse is the side length of the rhombus. Step 2: Determine the lengths of the half-diagonals. Given diagonals are d_1 = 12 cm and d_2 = 16 cm. The half-diagonals are: (d_1)/(2) = (12)/(2) = 6 cm (d_2)/(2) = (16)/(2) = 8 cm Step 3: Use the Pythagorean theorem to find the side length of the rhombus. Let s be the side length of the rhombus. In a right-angled triangle formed by the half-diagonals and a side, the Pythagorean theorem states a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. Here, a = 6 cm, b = 8 cm, and c = s. s^2 = (6 cm)^2 + (8 cm)^2 s^2 = 36 cm^2 + 64 cm^2 s^2 = 100 cm^2 s = sqrt(100 cm)^2 s = 10 cm Step 4: Calculate the perimeter of the rhombus. The perimeter P of a rhombus is 4 times its side length s. P = 4s P = 4 × 10 cm P = 40 cm The perimeter of the rhombus is 40 cm.