In $\triangle A B C, \angle A$ is obtuse, $P B \perp A C$ and $Q C \perp A B$. Prove that:
(i) $\mathrm{AB} \times \mathrm{AQ}=\mathrm{AC} \times \mathrm{AP}$
(ii) $B C^{2}=(A C \times C P+A B \times B Q)$
Given: ΔABC where ∠BAC is obtuse. PB ⊥AC and QC⊥AB.
To prove: (i) $\mathrm{AB} \times \mathrm{AQ}=\mathrm{AC} \times \mathrm{AP}$ and (ii) $\mathrm{BC}^{2}=\mathrm{AC} \times \mathrm{CP}+\mathrm{AB} \times \mathrm{BQ}$
Proof: In $\triangle \mathrm{ACQ}$ and $\triangle \mathrm{ABP}$,
$\angle C A Q=\angle B A P($ Vertically opposite angles $)$
$\angle Q=\angle P\left(=90^{\circ}\right)$
$\therefore \triangle \mathrm{ACQ} \sim \triangle \mathrm{ABP}[\mathrm{AA}$ similarity test]
$\Rightarrow \frac{\mathrm{CQ}}{\mathrm{BP}}=\frac{\mathrm{AC}}{\mathrm{AB}}=\frac{\mathrm{AQ}}{\mathrm{AP}}$ [Corresponding sides are in the same proportion]
$\frac{\mathrm{AC}}{\mathrm{AB}}=\frac{\mathrm{AQ}}{\mathrm{AP}}$
$\Rightarrow \mathrm{AQ} \times \mathrm{AB}=\mathrm{AC} \times \mathrm{AP}$.....(1)
In right $\triangle B C Q$,
$\Rightarrow B C^{2}=C Q^{2}+Q B^{2}$
$\Rightarrow B C^{2}=C Q^{2}+(Q A+A B)^{2}$
$\Rightarrow B C^{2}=C Q^{2}+Q A^{2}+A B^{2}+2 Q A \times A B$
$\Rightarrow B C^{2}=A C^{2}+A B^{2}+Q A \times A B+Q A \times A B$ [In right $\triangle \mathrm{ACQ}, \mathrm{CQ}^{2}+\mathrm{QA}^{2}=\mathrm{AC}^{2}$ ]
$\Rightarrow B C^{2}=A C^{2}+A B^{2}+Q A \times A B+A C \times A P$(Using (1))
$\Rightarrow B C^{2}=A C(A C+A P)+A B(A B+Q A)$
$\Rightarrow B C^{2}=A C \times C P+A B \times B Q$
- 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,
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,