Coordinate Geometry: SSC CGL Quantitative Aptitude

Formulas (3)

• Distance between two points · Distance formula
  $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad d_{\text{origin}} = \sqrt{x^2 + y^2}$$
• Midpoint, section formula and centroid · Midpoint, section point and centroid
  $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \qquad P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right) \qquad G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$$
• Area of a triangle from its corners · Area of a triangle from coordinates
  $$\text{Area} = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$$

Watch out for (7)

• Losing the minus sign when a coordinate is negative
  → Distance between two points
• Adding the gaps instead of squaring them
  → Distance between two points
• Dividing by 2 for a centroid
  → Midpoint, section formula and centroid
• Taking the centroid one third of the way from the corner
  → Midpoint, section formula and centroid
• Using the midpoint of the hypotenuse for any triangle
  → Midpoint, section formula and centroid
• Forgetting the half
  → Area of a triangle from its corners
• Reporting a negative area
  → Area of a triangle from its corners

Formulas (3)

• Slope of a line · Slope from two points and from an equation
  $$m = \frac{y_2 - y_1}{x_2 - x_1} \qquad y = mx + c \qquad ax + by + c = 0 \Rightarrow m = -\frac{a}{b}$$
• Where a line cuts the axes · Intercept form of a line
  $$\frac{x}{p} + \frac{y}{q} = 1$$
• Equation of a line from a point and its slope · Point-slope form
  $$y - y_1 = m(x - x_1)$$

Watch out for (6)

• Dropping the minus sign in minus a over b
  → Slope of a line
• Subtracting in a different order on top and bottom
  → Slope of a line
• Putting the wrong coordinate to zero
  → Where a line cuts the axes
• Stopping at the first option that fits
  → Testing a point on a line, and finding an unknown
• Swapping the coordinates when substituting
  → Testing a point on a line, and finding an unknown
• Keeping the wrong sign inside the bracket
  → Equation of a line from a point and its slope

Formulas (2)

• Slopes of parallel and perpendicular lines · Parallel and perpendicular conditions
  $$m_1 = m_2 \qquad m_1 m_2 = -1 \qquad \frac{a_1}{a_2} = \frac{b_1}{b_2} \qquad a_1a_2 + b_1b_2 = 0$$
• Shortest distance from a point to a line · Distance from a point to a line
  $$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}} \qquad d_{\text{parallel}} = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$$

Watch out for (6)

• Taking only the reciprocal, or only the negative
  → Slopes of parallel and perpendicular lines
• Keeping both roots when the unknown is squared
  → Slopes of parallel and perpendicular lines
• Drawing the bisector through an end instead of the midpoint
  → Perpendicular bisector of a segment
• Giving the bisector the segment's own slope
  → Perpendicular bisector of a segment
• Leaving out the square root on the bottom
  → Shortest distance from a point to a line
• Using unequal coefficients for parallel lines
  → Shortest distance from a point to a line