show that the quadrilateral whose vertices are A(3,0) B(4,5) C(-1,4) and D(-2,-1) taken in order is a rhombus hence find its area
Answers
Answer :-
checking length of all sides we get,
→ AB = √[(4 - 3)² + (5 - 0)²] = √(1 + 25) = √26
→ BC = √[(-1 - 4)² + (4 - 5)²] = √(25 + 1) = √26
→ CD = √[(-2 + 1)² + (-1 - 4)²] = √(1 + 25) = √26
→ DA = √[(3 + 2)² + (0 + 1)²] = √(25 + 1) = √26
now, checking lengths of diagonals we get,
→ AC = √[(-1 - 3)² + (4 - 0)²] = √(16 + 16) = √32 = 4√2
→ BD = √[(-2 - 4)² + (-1 - 5)²] = √(36 + 36) = √72 = 6√2
As we can see that, all sides are equal and diagonals are not equal . given quadrilateral is a rhombus .
therefore,
→ Area ABCD = (1/2) * D1 * D2 = (1/2) * 4√2 * 6√2 = (24 * 2)/2 = 24 units² (Ans.)
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