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
The diagonals of a parallelogram bisect each other.
Given: ABCD is a parallelogram.
To Prove: AO = OC and BO = OD.
Construction: Draw diagonal AC and diagonal BD.
ABCD is a parallelogram.
∴ AB || DC and AD || BC
AB || DC, AC is the transversal intersecting them at A and C respectively.
∠BAC = ∠DCA ........(Alternate interior angles)
Thus, ∠BAO = ∠ DCO .........(1)
As, AB || DC and BD is the transversal intersecting them at B and D respectively.
∠ABD = ∠CDB ........(Alternate interior angles)
So, ∠ABO = ∠CDO ...(2)
In △ AOB and △ COD,
∠ DCO ≅ ∠BAO .....(Pair of alternate interior angles)(1)
∠ CDO ≅ ∠ABO .....(Pair of alternate interior angles)(2)
side DC = side AB ....(parallel side)
by ASA congruency condition,
△ AOB ≅ △ COD.
This gives, AO = CO and BO = DO.....(C.P.C.T)
If OE = 4 then OP also is 4. .......(The diagonals of a parallelogram bisect each other.)
So, PE = 8,
Therefore, HL = 8 + 5 = 13
Hence, OH = `1/2 xx 13` = 6.5 cms.
Shaalaa.com | To Prove that the Diagonals of a Parallelogram Bisect Each Other
The point of intersection of diagonals of a quadrilateral divides one diagonal in the ratio 1:2. Can it be a parallelogram? Why or why not?
Two sticks each of length 5 cm are crossing each other such that they bisect each other. What shape is formed by joining their endpoints? Give reason.
Two sticks each of length 7 cm are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points? Give reason.
- Properties of a Parallelogram - Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Properties of a Parallelogram - Property: The Opposite Sides of a Parallelogram Are of Equal Length.