Geometry, Mensuration, Statistics & Coordinate Geometry
1. Triangles and Their Properties
Congruence Criteria (Review)
SSS. SAS. ASA. AAS. RHS (Right-Hypotenuse-Side).
Mid-Point Theorem
The line segment joining the MIDPOINTS of two sides of a triangle is PARALLEL to the third side and EQUAL TO HALF of it. 'If D and E are midpoints of AB and AC, then DE ∥ BC and DE = ½BC.'
Converse
A line drawn through the midpoint of one side, parallel to another side, BISECTS the third side.
Pythagoras Theorem
In a right triangle: (Hypotenuse)² = Sum of squares of other two sides. a² + b² = c² (c = hypotenuse).
2. Rectilinear Figures (Quadrilaterals)
| Shape | Key Properties |
|---|---|
| Parallelogram | Opposite sides ∥ and =. Opposite angles =. Diagonals BISECT each other. |
| Rectangle | Parallelogram + Angles = 90°. Diagonals =. |
| Rhombus | Parallelogram + All sides =. Diagonals ⟂, bisect angles. |
| Square | Rectangle + Rhombus. Diagonals = and ⟂. |
| Trapezium | ONE pair of parallel sides. |
3. Circle Theorems
- Angle at CENTRE = 2 × Angle at CIRCUMFERENCE (subtended by same arc).
- Angle in a SEMICIRCLE = 90°.
- Angles in the SAME SEGMENT are EQUAL.
- Equal CHORDS are EQUIDISTANT from centre.
- The PERPENDICULAR from centre to a chord BISECTS the chord.
4. Mensuration — Surface Area and Volume of Solids
| Solid | Surface Area | Volume |
|---|---|---|
| Cuboid | 2(lb + bh + hl) | lbh |
| Cube | 6s² | s³ |
| Cylinder | 2πr(r + h) | πr²h |
| Cone | πr(r + l) [l = slant height = √(r²+h²)] | ⅓πr²h |
| Sphere | 4πr² | (4/3)πr³ |
| Hemisphere | 3πr² | (2/3)πr³ |
5. Statistics
Measures of Central Tendency
- Mean (Arithmetic Average) = Σx/n. For grouped data: Mean = Σfx/Σf.
- Median: The MIDDLE value when data is ORDERED.
- Mode: Most FREQUENT value.
Graphical Representation
- Histogram: For CONTINUOUS grouped data. Rectangles touch (no gap).
- Frequency Polygon: Connect midpoints of histogram tops.
- Ogive (Cumulative Frequency Curve): Plot cumulative frequencies.
Mean from Assumed Mean (Shortcut Method)
Mean = A + (Σfd/Σf). Where A = assumed mean. d = x — A.
6. Coordinate Geometry
The Cartesian Plane
Ordered pair (x, y). x = abscissa. y = ordinate.
Distance Between Two Points
d = √[(x₂ — x₁)² + (y₂ — y₁)²]
Section Formula
Point dividing P and Q in ratio m:n internally: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
Slope of a Line
m = (y₂ — y₁)/(x₂ — x₁). Slope = tan θ. Positive slope = RISING. Negative = FALLING. Zero = HORIZONTAL.
Equation of a Line
- Slope-intercept form: y = mx + c.
- Point-slope: y — y₁ = m(x — x₁).
- Two-point: Use slope formula, then point-slope.
