Geometry: Distance and Midpoints

Distance between Points

• The distance between two points P (x1, y1) and Q (x2, y2) is calculated as : √((x2 - x1)^2 + (y2 - y1)^2)
• The distance between two points is the length of the line segment connecting them

Distance from Origin

• The distance of a point P (x, y) from the origin (0, 0) is calculated as √(x^2 + y^2)

Internal Division Point

• The coordinates of a point P that divides the line segment joining points A (x1, y1) and B (x2, y2) internally in the ratio m1:m2 are calculated as: ((m1 * x2 + m2 * x1) / (m1 + m2), (m1 * y2 + m2 * y1) / (m1 + m2))

Mid-point of Line Segment

• The coordinates of the mid-point of the line segment joining points P (x1, y1) and Q (x2, y2) are calculated as: ((x1 + x2)/2, (y1 + y2)/2)

Distance Between Specific Points

• The distance between the points (0, 5) and (-5, 0) is 5√2

Rectangle Properties

• AOBC, with vertices A (0, 3), O (0, 0), and B (5, 0), is a rectangle.
• The length of its diagonal is calculated using the Pythagorean theorem: √(AB)^2 + (OB)^2 = √5^2 + 3^2 = √34

Triangle Properties

• The perimeter of a triangle with vertices (0, 4), (0, 0), and (3, 0) is the sum of all its sides: 4 + 3 + √(3^2 + 4^2) = 12
• The area of a triangle with vertices A (3, 0), B (7, 0), and C (8, 4) is calculated using the formula: (1/2) * base * height = (1/2) * 4 * 4 = 8

Distance Between Two Points

• The distance between P ( x1 , y1 ) and Q ( x2 , y2 ) is ( ) ( )2 2 2 1 2 1– –x x y y+
• The distance of a point P (x,y) from the origin is 2 2 x y+

Section Formula

• The coordinates of the point P which divides the line segment joining the points A ( x1 , y1 ) and B ( x2 , y2 ) internally in the ratio m1 : m2 are 1 2 2 1 1 2 2 1 1 2 1 2 + + , + + m x m x m y m y m m m m      

Mid-point Formula

• The coordinates of the mid-point of the line segment joining the points P (x1, y1) and Q (x2, y2) are ( 1 2 2 1+ x x , 1 2 2 1+ y y )

Example 1

• The distance between the points (0, 5) and (–5, 0) is (B) 5 2

Example 2

• AOBC is a rectangle with vertices A (0, 3), O (0, 0), and B (5, 0).
• The length of its diagonal is (C) 34.

Example 3

• The perimeter of a triangle with vertices (0, 4), (0, 0), and (3, 0) is (C) 11

Example 4

• The area of a triangle with vertices A (3, 0), B (7, 0), and C (8, 4) is (A) 14