Geometry

Geometry

Concepts

• Vertex :- Point of intersection of two line segments or lines or rays is called Vertex. 

• Acute Angle - angle measures between 0° to 90°.

• Right Angle  - angle measures exactly 90°.

• Obtuse Angle  - angle measures between 90° to 180°.

• Reflex Angle - an angle greater than 180°, but less than 360°.

• Straight Angle - an angle equal to 180°.

• Complete Angle- An angle whose measure is 360°.

• Sum of all the measures of the angles around a point is 360°.

• Complementary Angle  - sum of two angles is equal to 90°.

• Supplementary Angle  - sum of two angles is equal to 180°.

• Two angles are adjacent, if they have a common vertex.

• If two lines intersect each other, then the vertically opposite angles are equal. 

• 1° = 60 minutes,  written as 60'

• 1' = 60 seconds,  written as 60"

Triangle 

Let a, b, c be the sides of △ABC, with a ≤ b ≤ c

1. a² + b² = c² → c is a right angle , so △ABC is a right angled triangle. 
2. a² + b² > c² → c is an acute angle , so △ABC is an acute angled triangle. 
3. a² + b² < c² → c is an obtuse angle , so △ABC is an abtuse angled triangle. 

• The measure of an exterior angle of a triangle is equal to sum of the measurement of two opposite interior angles.
• An attitude divides an equilateral triangle into two right angled triangles.

Triangle Inequality 

1. Sum of the lengths of any two sides of a triangle is greater than the length of the third side. i.e. x + y > z
2. Difference of the lengths of any two sides of a triangle is less than the length of the third side. i.e. x - y < z

Quadrilateral 

1. Rectangle

• Opposite sides are equal 
• Each angle 90°
• Diagonals are equal
• Area = length × breadth 
• Perimeter = 2 ×( length + breadth)

2. Square 

• All sides are equal 
• Each angle is 90°
• Diagonals are equal 
• Area = side² = diagonal²/2
• Diagonal = √2 × side
• Perimeter = 4 × side

3. Parallelogram 

• Opposite sides are parallel 
• Opposite sides are equal 
• Opposite angles are equal 
• Each diagonal bisect with each other 
• Diagonals are not equal 
• Sum of consecutive angles is 180°. 
• A diagonal divides into two triangles that have the exact same size and shape. i.e. triangles are congruent. 
• Area = base × height 
• Perimeter = sum of all sides
• The bisectors of angles form a rectangle. 

4. Rhombus 

• A Parallelogram having all sides are equal 
• Diagonals bisect each other at right angles 
• Area = 1/2 × (product of diagonals)
• Height = Area/side

5. Trapezium

• A quadrilateral in which one pair of sides is parallel and other pair of sides is not parallel. 
• Parallel sides are called bases. 
• Two bases are not equal. 
• Area = A= 1/2 × height × sum of parallel sides

    Height- distance between two parallel sides

• Median = 1/2 × sum of parallel sides 

👉 For a given perimeter, the rectangle with largest area is square. 

👉 For a given area, the rectangle with smallest perimeter is square. 

Circle

Let ‘r’ be the radius and ‘d’ be the diameter of the circle.

1. Area = πr² = πd²/4

2. Circumference = 2πr = πd

3. Length of an arc = 2πr θ/360, where θ is the central angle.

4. Area of the Sector = πr² θ/360

5. Area of a semi-circle = πr²/2

6. Circumference of a semi-circle = πr + 2r = r (π + 2)

7. Total angle covered by the Circle = 360°

• Radius of incircle of an equilateral triangle of side 'a' = a/2√3
• Radius of circumcircle of an equilateral triangle of side 'a' = a/√3

Some Important Points Regarding Circles :

• If two circles touch internally,  then the distance between their centres is equal to the difference of their radii.
• If two circles touch externally, then the distance between their centres is equal to the sum of their radii.
• Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel. 
• The number of revolutions completed by a rotating wheel in one minute 

  =(distance moved in one minute)/circumference

Polygon

• Convex Polygon - If none of the interier angles of a polygon is more than 180°, then it is called a convex polygon.

• Concave Polygon - If at least one angle of a polygon is more than 180°, then it is called a concave polygon.

• No. of diagonals of a polygon of n-sides = [{n(n - 1)/2} - n]

• An n-sides polygon is divided into (n - 2) triangles.

• The sum of measures of the 'n' angles in a polygon with n-sides is (n - 2) × 180°.

• In any polygon, the sum of the exterier angles, taking one ast each vertex is 360°.

• Regular Polygon - a polygon having all sides are same length and each angle has the same measure.

• In any regular polygon, the measure of each interier angle is (n - 2) × 180°/n and the measure of each exterier angle is 360°/n.

• Area of regular polygon = 1/2 (no. of sides) (radius of inscribed circle) = 3√3/2 ×(side)²

Some Useful Points 

1. Incenter 

• Point where the angle bisectors of a triangle meet.
• Center of incircle is Incenter.
• Lies always inside the triangle. 

2. Circumcenter

• Point where perpendicular bisectors of a triangle intersect. 
• Center of circumcircle is circumcenter. 
• Circumcenter is the intersection point of all midpoints.

3. Orthocenter 

• Point located at intersection of altitudes of a triangle. 
• It is not always inside the triangle. 
• If the triangle is abtuse, Orthocenter will be at outside.

4. Median - It is a line segment that joins a vertex to the midpoint of the side that is opposite to the vertex. 

5. Centroid  - The point at which the three medians of the triangle intersect is known as centroid of a triangle. 

• It is also known as point of intersection of all three medians. 
• The centroid divides each of median in the ratio of 2 : 1.

👉 Orthocenter, Centroid and Circumcenter of any triangle are colinear, i.e. they always lie on the same straight line.

• The Orthocenter of a right angled triangle lies at the right angular vertex.

• If in a triangle,  the circumcenter, incenter, centroid and orthocenter coincide,  then the triangle is equilateral triangle. 

• The equidistant point from the vertices of a triangle is called circumcenter. 

Examples 

1. The area of the largest triangle that can be inscribed in a semi-circle of radius r is ---?
Solution:
 Area of triangle = 1/2 × b × h = 1/2 × 2r × r = r²

2. The area of a circle is 220 cm². The area of a square inscribed in the circle is ----?
Solution:
Given πr² = 220, then  r² = 70
Here, r = diagonal/2
Area of square = 1/2 × (diagonal)² = 140

3. The radius of circle is 13 cm and AB is a chord which is at a distance of 12 cm from the centre. The length of the chord is :----?
Solution:

4. A chord of length 30 cm is at a distance of 8 cm from the centre of a circle. The radius of the circle is ----?
Solution:

5. Two supplementary angles are in the ratio 3 : 2. The smaller angle measures---? 
Solution:

Previous Year Questions

1. An equilateral triangle of side 6 cm is inscribed in a circle.  The radius of the circle is ---? (Ans. 2√3 cm)

2. A right angled triangle has a height a = 138 cm, base b = 184 cm and hypotenuse c. Find the value of c ? (Ans. 230 cm)

3. The diameter of a circle is 28 cm. What is the length of its circumference? (Ans. 88 cm)

4. If D is a point on side BC of a triangle ABC such that AD = BD = CD, then what is relationship between the sides of the triangle? (Ans. AB² + AC² = BC²)

5. A diagonal of a rectangle is inclined to one side of the rectangle at 35°. The acute angle between the diagonals is---? (Ans.70°)

6. PQR is a triangle in which ∠P = 84°, if the internal bisectors of ∠Q and ∠R meet at O, then measure of ∠QOR is -----? (Ans. 132°)

7. ABCD is a cyclic quadrilateral whose side AB is the diameter.  If ∠ADC = 120°, then ∠BAC = ? (Ans. 30°)

8. A rectangle ABCD has dimensions of 20 cm and 30 cm. What is the maximum number of circles of radius 1 cm can be drawn within the rectangle without intersecting each other?(Ans. 150)

9. A chord of length 16 cm is drawn in a circle of radius 10 cm. The distance of the chord from the centre of the circle is :-----? (Ans. 6 cm)

10. A ladder is placed in such way that its foot is 15 m away from the wall and its top reaches a window 20 m above the ground. The length of the ladder is :---? (Ans. 25 m) 

11. The perimeters of two similar triangles △ABC  and △PQR are 36 cm and 24 cm respectively. If PQ = 10 cm, then AB =---? (Ans. 15 cm)