The points \(A(1,\,1)\), \(B(4,\,5)\), \(C(8,\,2)\) and \(D(5,\,-2)\) are joined in order to form the quadrilateral ABCD. Which description fits ABCD?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → a square
A quadrilateral with all four sides equal and both diagonals equal is a square. (Four equal sides alone give a rhombus; equal diagonals as well make it a square.)
\(AB^2=(4-1)^2+(5-1)^2=9+16=25\), so \(AB=5\)
\(BC^2=(8-4)^2+(2-5)^2=16+9=25\), so \(BC=5\)
\(CD^2=(5-8)^2+(-2-2)^2=9+16=25\), so \(CD=5\)
\(DA^2=(1-5)^2+(1+2)^2=16+9=25\), so \(DA=5\)
Diagonals: \(AC^2=(8-1)^2+(2-1)^2=49+1=50\) and \(BD^2=(5-4)^2+(-2-5)^2=1+49=50\), so \(AC=BD=5\sqrt{2}\)
All sides equal and diagonals equal, so ABCD is a square
