Linear Equations in Two Variables
Introduction to Euclid’S Geometry
Lines and Angles
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
- Types of Quadrilaterals
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Angle Subtended by a Chord at a Point
- Perpendicular from the Centre to a Chord
- Circles Passing Through One, Two, Three Points
- Equal Chords and Their Distances from the Centre
- Angle Subtended by an Arc of a Circle
- Cyclic Quadrilateral
Areas - Heron’S Formula
Surface Areas and Volumes
Statistics and Probability
Heron was born in about 10AD possibly in Alexandria in Egypt. He worked in applied mathematics. His geometrical works deal largely with problems on mensuration written in three books. In this book, Heron has derived the famous formula for the area of a triangle in terms of its three sides.
The formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as:
|Area of a triangle = `sqrt (s(s-a) (s - b)(s-c))`|
where a, b and c are the sides of the triangle, and s = semi-perimeter, i.e., half the perimeter of the triangle = `(a + b + c) /2`,
For instances, Let us take a = 40 m, b = 24 m, c = 32 m,
so that we have s = `(40 + 24 + 32 )/ 2` m
= 48 m
s - a = (48 - 40) m = 8m
s - b = (48 - 24)m = 24 m
s - c = (48 - 32 ) m = 16 m
Therefore, area of the park ABC
`= sqrt (s(s - a) (s - b) (s - c))`
`= sqrt (48 * 8 * 24 * 16) m^2 = 384 m^2`
- Collinearity of three points
Shaalaa.com | Area of a Triangle — by Heron’s Formula
A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 13 cm, 14 cm and 15 cm and the parallelogram stands on the base 14 cm, find the height of the parallelogram.
The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side.
Find the area of a quadrilateral ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA =34 cm and diagonal BD =20 cm.
The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.
The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of the parallelogram.