# Mission IIMpossible - Gyan Room (Geometry - Short cuts/Questions/Concepts)

This thread is reserved for sharing concepts, short cuts and good questions from Geometry topic.

Rules:

1. Mention source (if required)
2. Provide examples for short-cuts
3. Posts not related to Geometry will be removed.

• (a) In a plane if there are n points of which no three are collinear, then
•The number of straight lines that can be formed by joining them is nC2.
•The number of triangles that can be formed by joining them is nC3.
•The number of polygons with k sides that can be formed by joining them is nCk.

(b) In a plane if there are n points out of which m points are collinear, then
•The number of straight lines that can be formed by joining them is nC2 – mC2 + 1.
•The number of triangles that can be formed by joining them is nC3 – mC3.
•The number of polygons with k sides that can be formed by joining them is nCk – mCk.

(c) The number of diagonals of a n sided polygon are nC2 – n = n × (n – 3)/2.

(d) The number of triangles that can be formed by joining the vertices of a n-sided polygon which has,
•Exactly one side common with that of the polygon are n × (n – 4).
•Exactly two sides common with that of the polygon are n.
•No side common with that of the polygon are n × (n – 4) × (n – 5)/6.

(e) The number of parallelograms formed if ‘x’ lines in a plane are intersected by ‘y’ parallel lines are x × y × (x – 1) × (y – 1)/4.

(f) If there are n lines drawn in a plane such that no two of them are parallel and no three of them are concurrent, then the number of different points at which these lines will intersect each other is nC2 = n × (n – 1)/2.

(g) If there are n straight lines in a plane and no two of them are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of new lines obtained are n × (n – 1) × (n – 2) × (n – 3)/8.

(h) There are three co-planar parallel lines. If p points are taken on each of the lines, the maximum number of triangles with vertices at these points is p3 + 3p2(p – 1).
(i) The sides of a triangle a, b and c are integers where a ≤ b ≤ c. If c is given then the number of different triangles is c × (c + 2)/4 or (c + 1)2/4, according to c as even or odd. Also, the number of isosceles triangles is (3c – 2)/2 or (3c – 1)/2, according to c as even or odd.

(j) In a square of n x n,
•The number of rectangles of any size is ∑r³.
•The number of squares of any size is ∑r².

(k) In a rectangle of p x q (p < q),
•The number of rectangles of any size is p x q x (p + 1) x (q + 1) / 4.
•The number of squares of any size is ∑ [(p + 1 – r) x (q + 1 – r)].

(l) If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of parts into which these lines divide the plane is equal to [n x (n + 1)/2] + 1.

(m) If “n” parallel lines are passing through a circle, dividing the plane into distinct non-overlapping bounded or unbounded regions, then the maximum number of regions into which the plane can be divided is (3n + 1).

• In an isosceles right angled triangle , both the acute angles are equal to 45°
The hypotenuse = √2 × perpendicular side.
For a right angled triangle of given perimeter, an isosceles right angled triangle has maximum area.
A square is broken into two isosceles right angled triangles by its diagonal.
Ratio of sides : 1:1 :√2.

Right angled triangle with angles 30°- 60°- 90°
The side opposite to 30° = 1/2 × hypotenuse
The side opposite to 60° = √3/2 × hypotenuse
In an 30°- 60°- 90° triangle , the measure of the sides are in the ratio : 1:√3:2
One half of an equilateral triangle is a 30°- 60°- 90° triangle.

• Here r = Radius of circle
a = Side of square
n = Number of circles

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