Mathematics: 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.

648Step-by-step explanation:...

Determine whether parallelogram JKLM with vertices

Ответ:

Jazongamez1987

18.01.2022

Mathematics

JKLM represents all three(Rhombus, Square and Rectangle)

Step-by-step 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:

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

$all\ sides\ are\ equal\\\\Now\ find\ out\ diagonal\\\\JL=\sqrt{(4+1)^2+(6-1)^2}\\\\JL=\sqrt{5^2+5^2}\\\\JL=\sqrt{25+25}\\\\JL=\sqrt{50}\\\\JL=\sqrt{25\times 2}\\\\JL=5\sqrt{2}\\\\KM=\sqrt{(-1-4)^2+(6-1)^2}\\\\KM=\sqrt{(-5)^2+(5)^2}\\\\KM=\sqrt{25+25}\\\\KM=\sqrt{50}\\\\KM=\sqrt{25\times 2}\\\\KM=5\sqrt{2}$

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

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

Determine whether parallelogram JKLM with vertices J(-1, 1), K(4, 1), L(4, 6) and M(-1, 6) is a rhom

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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.

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