Name the quadrilateral ABCD, the coordinates of whose vertices are A(3

1.	Name the quadrilateral ABCD, the coordinates of whose vertices are A(3, 5), B(1, 1), C(5, 3) and D(7, 7).(a) Square (b) Rhombus (c) Rectangle (d) Trapezium
Answer» (b) Rhombus AB = \(\sqrt{(3-1)^2+(5-1)^2}\) = \(\sqrt{4+16}\) = \(\sqrt{20}\) = \(2\sqrt5\) BC = \(\sqrt{(1-5)^2+(1-3)^2}\) = \(\sqrt{16+4}\) = \(\sqrt{20}\) = \(2\sqrt5\) CD = \(\sqrt{(5-7)^2+(3-7)^2}\) = \(\sqrt{4+16}\) = \(\sqrt{20}\) = \(2\sqrt5\) AD = \(\sqrt{(3-7)^2+(5-7)^2}\) = \(\sqrt{16+4}\) = \(\sqrt{20}\) = \(2\sqrt5\) AC = \(\sqrt{(3-5)^2+(5-3)^2}\) = \(\sqrt{4+4}\) = \(\sqrt{8}\) = \(2\sqrt2\) BD = \(\sqrt{(1-7)^2+(1-7)^2}\) = \(\sqrt{36+36}\) = \(\sqrt{72}\) = \(6\sqrt2\) Now, AB = BC = CD = AD ⇒ All sides are equal Also, AC ≠ BD ⇒ Diagonals are not equal. ⇒ ABCD is a rhombus.

Name the quadrilateral ABCD, the coordinates of whose vertices are A(3, 5), B(1, 1), C(5, 3) and D(7, 7).(a) Square (b) Rhombus (c) Rectangle (d) Trapezium

Answer»

(b) Rhombus

AB = \(\sqrt{(3-1)^2+(5-1)^2}\) = \(\sqrt{4+16}\) = \(\sqrt{20}\) = \(2\sqrt5\)

BC = \(\sqrt{(1-5)^2+(1-3)^2}\) = \(\sqrt{16+4}\) = \(\sqrt{20}\) = \(2\sqrt5\)

CD = \(\sqrt{(5-7)^2+(3-7)^2}\) = \(\sqrt{4+16}\) = \(\sqrt{20}\) = \(2\sqrt5\)

AD = \(\sqrt{(3-7)^2+(5-7)^2}\) = \(\sqrt{16+4}\) = \(\sqrt{20}\) = \(2\sqrt5\)

AC = \(\sqrt{(3-5)^2+(5-3)^2}\) = \(\sqrt{4+4}\) = \(\sqrt{8}\) = \(2\sqrt2\)

BD = \(\sqrt{(1-7)^2+(1-7)^2}\) = \(\sqrt{36+36}\) = \(\sqrt{72}\) = \(6\sqrt2\)

Now, AB = BC = CD = AD ⇒ All sides are equal

Also, AC ≠ BD ⇒ Diagonals are not equal.

⇒ ABCD is a rhombus.

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