Question

prove that :

The distance between Two points is given by the distance formula ​

Answer 1

Let d = distance between the points A and B.

Draw AL and BM perpendicular to x-axis (parallel to y-axis).

Draw AC perpendicular to BM to cut BM at C.

In the figure,

OL = x1x1 , OM = x2x2

Thus, AC = LM = OM - OL = x2−x1x2−x1

MB = y2y2 , MC = LA = y1y1

So, CB = MB - MC = y2−y1y2−y1

From the right-angled triangle ACB,

AB2=AC2+CB2AB2=AC2+CB2

d2=(x2−x1)2+(y2−y1)2d2=(x2−x1)2+(y2−y1)2

∴ d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√∴ d=(x2−x1)2+(y2−y1)2

d=(Difference between abcissae)2+(Difference between ordinates)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√d=(Difference between abcissae)2+(Difference between ordinates)2

If the points A and B lie on the X-axis, then the ordinates of A and B are zeros.

i.e., A (x 11 , 0), B (x 22 , 0)

AB = (x2−x1)2−−−−−−−−√(x2−x1)2 = |x2−x1||x2−x1|

If the points A and B lie on the y-axis, then the abscissae of A and B are zeros.

i.e., A (0, y 11 ) and B (0, y 22 )

AB = (y2−y1)2−−−−−−−−√(y2−y1)2 = |y2−y1||y2−y1|

Distance of any point A (x, y) from the origin

OA = d = (x−0)2+(y−0)2−−−−−−−−−−−−−−−√(x−0)2+(y−0)2 = x2+y2−−−−−−√

Answer 2

Answer:

Answer in Attachment bro