$\triangle \mathrm{ABD}$ is a right triangle right-angled at $\mathrm{A}$ and $\mathrm{AC} \perp \mathrm{BD}$. Show that
(i) $\mathrm{AB}^{2}=\mathrm{BC} \cdot \mathrm{BD}$
(ii) $\mathrm{AC}^{2}=\mathrm{BC} . \mathrm{DC}$
(iii) $\mathrm{AD}^{2}=\mathrm{BD} \cdot \mathrm{CD}$
(iv) $\frac{\mathrm{AB}^{2}}{\mathrm{AC}^{2}}=\frac{\mathrm{BD}}{\mathrm{DC}}$
In $\triangle A B D$ and $\triangle A B C$,
$\angle A C B=\angle A=90^{\circ}$
$\angle B=\angle B$ (Common angle)
So, by AA criterion $\triangle A B D \sim \triangle C B A$
$\therefore \frac{A B}{B C}=\frac{B D}{A B}=\frac{A D}{A C}$
$\therefore \frac{A B}{B C}=\frac{B D}{A B}$
$\therefore A B^{2}=B D \cdot B C$....(1)
(ii) In $\triangle A B D$ and $\triangle A C D$,
$\angle C=\angle A=90^{\circ}$
$\angle D=\angle D \quad$ (Common angle)
So, by AA criterion $\triangle A B D \sim \triangle C A D$
$\therefore \frac{A B}{A C}=\frac{B D}{A D}=\frac{A D}{C D}$
$\therefore \frac{B D}{A D}=\frac{A D}{C D}$
$\therefore A D^{2}=B D \cdot C D$....(2)
(iii) We have shown that $\triangle A B D$ is similar to $\triangle C B A$ and $\triangle A B D$ is similar to $\triangle C A D$ therefore, by the property of transitivity $\triangle C B A$ is similar to $\triangle C A D$.
$\therefore \frac{B C}{A C}=\frac{A B}{A D}=\frac{A C}{C D}$
$\therefore \frac{B C}{A C}=\frac{A C}{C D}$
$\therefore A C^{2}=B C \cdot C D$....(3)
(iv) Now to obtained $\mathrm{AB}^{2} / \mathrm{AC}^{2}=\mathrm{BD} / \mathrm{DC}$, we will divide equation (1) by equation (2) as shown below,
$\therefore \frac{A B^{2}}{A C^{2}}=\frac{B D \cdot B C}{B C \cdot C D}$
Canceling BC we get,
$\frac{A B^{2}}{A C^{2}}=\frac{B D}{C D}$
Therefore, $\frac{A B^{2}}{A C^{2}}=\frac{B D}{C D}$
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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,