Using slopes. Prove that the points A(-2, -1), B(1,0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.
The property of parallelogram states that opposite sides are equal.
We have 4 sides as AB,BC,CD,DA
Given points are A(-2,-1),B(1,0),C(4,3) and D(1,2)
AB and CD are opposite sides, and BC and DA are the other two opposite sides.
So slopes of AB = CD and slopes BC = DA
slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$
Slope of $A B=\left(\frac{0+1}{1+2}\right)=\frac{1}{3}$
The slope of $B C=\left(\frac{3-0}{4-1}\right)=\frac{3}{3}=1$
The slope of $C D=\left(\frac{2-3}{1-4}\right)=\frac{-1}{-3}=\frac{1}{3}$
Slope of $D A=\left(\frac{2+1}{1+2}\right)=\frac{3}{3}=1$
Therefore the Slope of $A B=$ Slope of $C D$ and
The slope of $B C=$ Slope of $D A$
Also, the product of slope of two adjacent sides is not equal to $-1$, therefore it is not a rectangle.
Hence ABCD is a parallelogram.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,