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Yogita Ingle 5 years, 4 months ago
Triangles are a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180o.
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Meghna Thapar 5 years, 4 months ago
Geometry is the part of mathematics that studies the size, shapes, positions and dimensions of things. ... Squares, circles and triangles are some of the simplest shapes in flat geometry. Cubes, cylinders, cones and spheres are simple shapes in solid geometry. The definition of geometry is a branch of math that focuses on the measurement and relationship of lines, angles, surfaces, solids and points. An example of geometry is the calculation of a triangle's angles. Geometry helps us in deciding what materials to use, what design to make and also plays a vital role in the construction process itself. Different houses and buildings are built in different geometric shapes to give a new look as well as to provide proper ventilation inside the house.
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Gaurav Seth 5 years, 4 months ago
Let positive integer a = 4m + r , By division algorithm we know here 0 ≤ r < 4 , So
When r = 0
a = 4m
Squaring both side , we get
a2 = ( 4m )2
a2 = 4 ( 4m2)
a2 = 4 q , where q = 4m2
When r = 1
a = 4m + 1
squaring both side , we get
a2 = ( 4m + 1)2
a2 = 16m2 + 1 + 8m
a2 = 4 ( 4m2 + 2m ) + 1
a2 = 4q + 1 , where q = 4m2 + 2m
When r = 2
a = 4m + 2
Squaring both hand side , we get
a2 = ( 4m + 2 )2
a2 = 16m2 + 4 + 16m
a2 = 4 ( 4m2 + 4m + 1 )
a2 = 4q , Where q = 4m2 + 4m + 1
When r = 3
a = 4m + 3
Squaring both hand side , we get
a2 = ( 4m + 3)2
a2 = 16m2 + 9 + 24m
a2 = 16m2 + 24m + 8 + 1
a2 = 4 ( 4m2 + 6m + 2) + 1
a2 = 4q + 1 , where q = 4m2 + 6m + 2
Hence
Square of any positive integer is in form of 4q or 4q + 1 , where q is any integer.
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