Question

Show that the points of a (0,-1) b(-2,3) c(6,7) and d(8,3) are the vertices of a rectangle ABCD

Answer 1

Answer:

${ \large{ \pmb{ \sf{★Given... }}}}$

A(0, -1)

B(-2, 3)

C(6, 7)

D(8, 3)

${ \large{ \pmb{ \sf{★To \: Find... }}}}$

Show that points are vertices of rectangle..?

${ \large{ \pmb{ \sf{ ★Used \: Formula...}}}}$

${ \sf{Distance = \sqrt{ {( x_{2} -x_{1} )}^{2} + {( y_{2} - y_{1})}^{2} } }}$

${ \large{ \pmb{ \sf{★Solution... }}}}$

Sides:

${ \implies{ \sf{AB = \sqrt{ {( - 2 - 0)}^{2} + {(3 - ( - 1))}^{2} } }}}$

${ \implies{ \sf{AB = \sqrt{ {( - 2)}^{2} + {(3 + 1)}^{2} } }}}$

${ \implies{ \sf{AB = \sqrt{4 + 16} }}}$

${ \implies{ \sf{AB = \sqrt{20} }}}$

${ \implies{ \bf{AB = 2 \sqrt{5} }}}$

____________________________

${ \implies{ \sf{BC = \sqrt{ {(6 - ( - 2))}^{2} + {(7 - 3)}^{2} } }}}$

${ \implies{ \sf{BC = \sqrt{ {(6 + 2)}^{2} + {(4)}^{2} } }}}$

${ \implies{ \sf{BC = \sqrt{64 + 16} }}}$

${ \implies{ \bf{BC = \sqrt{70} }}}$

_____________________________

${ \implies{ \sf{CD = \sqrt{ {(8 - 6)}^{2} + {(3 - 7)}^{2} } }}}$

$\: { \implies{ \sf{CD = \sqrt{ {( - 2)}^{2} + {( - 4)}^{2} } }}}$

$\: { \implies{ \sf{CD = \sqrt{4 + 16} }}}$

$\: { \implies{ \sf{CD = \sqrt{20} }}}$

$\: { \implies{ \bf{CD = 2 \sqrt{5} }}}$

____________________________

${ \implies{ \sf{DA = \sqrt{ {(0 - 8)}^{2} + {( - 1 - 3)}^{2}} }}}$

$\: { \implies{ \sf{DA = \sqrt{ {(8)}^{2} + {(4)}^{2} } }}}$

$\: { \implies{ \sf{DA = \sqrt{64 + 16} }}}$

$\: { \implies{ \bf{DA = \sqrt{70} }}}$

____________________________

Diagnols:

${ \implies{ \sf{AC = \sqrt{ {(6 - 0)}^{2} + {(7 + 1)}^{2} } }}}$

$\: { \implies{ \sf{AC = \sqrt{ {(6)}^{2} + {(8)}^{2} } }}}$

$\: { \implies{ \sf{AC = \sqrt{36 + 64} }}}$

$\: { \implies{ \sf{AC = \sqrt{100} }}}$

$\: { \implies{ \bf{AC = 10}}}$

______________________________

${ \implies{ \sf{BD = \sqrt{ {(8 + 2)}^{2} + {(3 - 3)}^{2} } }}}$

$\: { \implies{ \sf{BD = \sqrt{ {(10)}^{2} + {(0)}^{2} } }}}$

$\: { \implies{ \sf{BD = \sqrt{100} }}}$

$\: { \implies{ \bf{BD = 10}}}$

______________________________

From this,

Opposite sides are equal AB = CD, BC = DA

Diangnols are equal AC = BD

${ \large{ \pmb{ \sf{★Final \: Answer... }}}}$

The given points form a Rectangle

These points are vertices of Rectangle