LEARN TESTS

Equations of Geometrical Figures

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Slope-intercept form:
The slope-intercept form of the equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept of the line.

The y-intercept of a line is the y-co-ordinate of the point at which the line intersects the y-axis. Likewise, the x-intercept of a line is the x-co-ordinate of the point at which the line intersects the x-axis. Therefore, if the slope-intercept form of the equation of a line is given, we can find both intercepts. For example, in order to find the y-intercept, simply set x = 0 and solve for the value of y. For the x-intercept, set y = 0 and solve for x.

If both the x-intercept and y-intercept of a line is given, then the equation of the line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.

Point-slope form: The point-slope form of the equation of a line is y – y1 = m(x – x1), where m is the slope of the line and (x1, y1) is a point on the line.

Two-point form:
The equation of a line passing through the points (x1, y1) and (x2, y2) is given by,

Angle between two lines

  1. If m1 and m2 are the slopes of two lines, the angle θ between them is given by,


  2. If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are equations of two lines, the angle θ between them is given by,
  3. If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are equations of two lines, then these two lines are parallel when,


  4. If a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are equations of two lines, then these two lines are perpendicular to each other when,
    a1a2 + b1b2 = 0

Condition for parallel lines

a1/a2 = b1/b2 or m1 = m2

Condition for perpendicular lines

a1a2 + b1b2 = 0 or m1m2 = -1

Area of the triangle
Area of the triangle formed by the vertices A(x1, y1), B(x2, y2), C(x3, y3) is given by,


If the area of the triangle is zero, then it indicates that the three points A, B and C are collinear i.e. lie on the same straight line.

Section formula
i. Internal Division:

2. External Division:


Equations of circle
The equation of a circle centred at (h, k) with radius ‘r’ units is: (x - h)2 + (y - k)2 = r2

The equation of a circle centred at the origin with radius ‘r’ units is: x2 + y2 = r2

Centroid of a triangle


Circumcentre of a triangle
OA2 = OB2 = OC2
(x - x1)2 + (y - y1)2 = (x - x2)2 + (y - y2)2 = (x - x3)2 + (y - y3)2
Where, A, B and C are the vertices of the triangle and O is the circumcenter.

Incentre of a triangle

 

 


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