Question

Determine whether parallelogram JKLM with vertices J(-1, 1), K(4, 1), L(4, 6) and M(-1, 6) is a rhombus, square, rectangle or all three.

Answer 1

JKLM represents all three(Rhombus, Square and Rectangle)

Explanation:

Concepts Used:

Pythagorean Theorem: In a right triangle the square of longest side is equal to the sum of the squares of the other sides.

Distance between two points:

`Distance\ between\ (x_1,y_1)(x_2,y_2)\ is\ given\ by\\\\d=√((x_2-x_1)^2+(y_2-y_1)^2)\\\\J(-1,1)\ K(4,1)\ L(4,6)\ M(-1,6)\\\\JK=√((4+1)^2+(1-1)^2)\\\\JK=√(5^2+0^2)\\\\JK=√(25)\\\\JK=5\\\\KL=√((4-4)^2+(6-1)^2)\\\\KL=√(0^2+5^2)\\\\KL=√(25)\\\\KL=5\\\\LM=√((-1-4)^2+(6-6)^2)\\\\LM=√((-5)^2+0^2)\\\\LM=√(25)\\\\LM=5\\\\JM=√((-1+1)^2+(6-1)^2)\\\\JM=√(0^2+5^2)\\\\JM=√(25)\\\\JM=5\\\\`

`all\ sides\ are\ equal\\\\Now\ find\ out\ diagonal\\\\JL=√((4+1)^2+(6-1)^2)\\\\JL=√(5^2+5^2)\\\\JL=√(25+25)\\\\JL=√(50)\\\\JL=√(25* 2)\\\\JL=5√(2)\\\\KM=√((-1-4)^2+(6-1)^2)\\\\KM=√((-5)^2+(5)^2)\\\\KM=√(25+25)\\\\KM=√(50)\\\\KM=√(25* 2)\\\\KM=5√(2)`

`Now\ in\ \triangle JKL\\\\JL^2=JK^2+KL^2\\\\Hence\ \triangle JKL\ is\ a\ right\ triangle\\\\and\ \angle K=90\textdegree\\\\Similarly\ in\ \triangle KLM\\\\KM^2=KL^2+LM^2\\\\Hence\ \triangle KLM\ is\ a\ right\ triangle\\\\\angle L=90\textdegree\\\\Similerly\ \triangle JML\\\\JL^2=JM^2+ML^2\\\\Pythagorean\ theorem\ satisfied\\\\It\ is\ a\ right\ triangle\\\\\angle M=90\textdegree\\\\\angle J=360-(\angle M+\angle L+\angle K)\\\\\angle J=360-(90+90+90)\\\\\angle J=90\textdegree\\\\`

`All\ sides\ are\ equal\ and\ angles\ are\ 90\textdegree\\\\Hence\ it\ is\ a\ square\\\\ Every\ square\ is\ a\ rectangle\ and\ Every\ square\ is\ a\ rhombus.\\\\Hence\ it\ represents\ all\ three`