ABCD is a parallelogram. E is a point on BA such that BE = 2EA and F is point on DC such that DF = 2FC.
Question:
ABCD is a parallelogram. E is a point on BA such that BE = 2EA and F is point on DC such that DF = 2FC. Prove that AECF is a parallelogram whose area is one third of the area of parallelogram ABCD.
Solution:
Draw FG ⊥ AB
We have,
BE = 2 EA and DF = 2FC
⇒ AB - AE = 2 AE and DC - FC = 2 FC
⇒ AB = 3 AE and DC = 3 FC
⇒ AE = (1/3) AB and FC = (1/3)DC ⋅⋅⋅⋅ (1)
But AB = DC
Then, AE = FC [opposite sides of ∥gm]
Thus, AE = FC and AE ∥ FC
Then, AECF is a parallelogram
Now, area of parallelogram AECF = AE × FG
⇒ ar(∥gm AECF) = 1/3 AB × FG from(1)
⇒ 3ar (∥gm AECF) = AB × FG ⋅⋅⋅ (2)
And ar(∥gm ABCD) = AB × FG ⋅⋅⋅ (3)
Compare equation 2 and 3
⇒ 3ar(∥gm AECF) = ar(∥gm ABCD)
⇒ ar(∥gm AECF) = 1/3 ar(∥gm ABCD)
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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,
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