Final answer:
The perimeter of rhombus PQRS with diagonals of lengths 8 and 12 is found by dividing each diagonal in half and using the Pythagorean theorem to find the length of a side. Multiplying the side length by 4 gives a perimeter of approximately 28.844 units.
Step-by-step explanation:
The student is asking how to find the perimeter of a rhombus with given diagonal lengths. For rhombus PQRS with diagonals PR = 8 and QS = 12, we can use the property that diagonals of a rhombus bisect each other at right angles. This property will help us form four right-angled triangles within the rhombus (two for each diagonal). We can then use the Pythagorean theorem to find the length of each side of the rhombus.
To use the Pythagorean theorem, first calculate the half lengths of the diagonals. Half of PR is 4 and half of QS is 6. In each right-angled triangle formed by these halves, the sides are of length 4 and 6 which correspond to the halves of the diagonals.
Applying the Pythagorean theorem:
- a² + b² = c²
- 4² + 6² = c²
- 16 + 36 = c²
- c² = 52
- c = √52
The length of each side of the rhombus (let's call it s) is √52. The perimeter (P) of the rhombus is all four sides added together.
P = 4s = 4(√52) = 4(7.211)
(em)approximately(em)
The perimeter is approximately 28.844 units.