ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQ is equal to the rectangle contained by AB and BC.
Given:
ABCD is a parallelogram and APQ is a straight line meeting BC at P and DC produced at Q.
To Prove:
The rectangle obtained by BP and DQ is equal to the rectangle contained by AB and BC. We need to prove that BP×DQ=AB×BC
Proof:
In ∆ABP and ∆QCP, we have
∠ABP = ∠QCP (Alternate angles as AB || DC)
∠BPA = ∠QPC ( Vertically opposite angles)
By AA similarity, we get
∆ABP ~ ∆QCP
We know that corresponding sides of similar triangles are proportional.
⇒ABQC=BPCP=APQP
⇒ABQC=BPCP⇒AB×CP=QC×BP
Adding AB×BP in both sides, we get
⇒AB×CP+AB×BP=QC×BP+AB×BP⇒AB×CP+BP=QC+AB×BP⇒AB×CP+BP=QC+CD×BP
ABCD is a parallelogram, AB = CD⇒AB×BC=DQ×BP⇒BP×DQ=AB×BC
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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,