Identify the polygon with vertices \( P(2,-2), Q(4,2) \), \( R(6,2) \), and \( S(4,-2) \), and then find the perimeter and area of the polygon. rhombus; \( P=16 \) units; \( A=21 \) units \( ^{2} \) parallelogram; \( P=(4+4 \sqrt{5}) \) units; \( A=8 \) units \( ^{2} \) rhombus; \( P=21 \) units; \( A=16 \) units \( ^{2} \) parallelogram; \( P=8 \) units; \( A=(4+4 \sqrt{5}) \) units 2
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The Deep Dive
Did you know that the polygon formed by the vertices \( P(2,-2) \), \( Q(4,2) \), \( R(6,2) \), and \( S(4,-2) \) is actually a rhombus? This is because all sides have equal lengths, and its diagonals bisect each other at right angles! The coordinates reveal that the opposite sides are both parallel and equal in length, characteristic of rhombuses. When it comes to finding the perimeter and area of this rhombus, the perimeter \( P \) is calculated by adding the lengths of all sides, which results in \( P = 16 \) units. For the area \( A \), you can use the formula \( A = \frac{1}{2} \times d_1 \times d_2 \) (where \( d_1 \) and \( d_2 \) are the diagonals), resulting in \( A = 21 \) square units. It's a delightful geometry exercise, perfect for impressing your math friends!
